Sharing Demographic Risk – Who is Afraid of the Baby Bust?

42 Pages·2008·0.39 MB·English

Sharing Demographic Risk – Who is Afraid of the
Baby Bust?∗
Alexander Ludwig† Michael Reiter‡
August 2008
Abstract
We model the optimal reaction of a public PAYG pension system to demographic
shocks. Wecomparetheex-antefirstbestandsecondbestsolutionofaRamseyplan-
ner with full commitment to the outcome under simple third best rules that mimic
the pension systems observed in the real world. Themodel, in particular the pension
system, is calibrated to the German economy. The objective of the social planner is
calibrated such that the size of the German pension system was optimal under the
economic and demographic conditions of the 1960s. We find that the German sys-
tem comes relatively close to the second-best solution, especially when labor market
distortions are correctly modelled. Furthermore, the German system and a constant
contribution rate lead to a lower variability of lifetime utility than does the second
best policy. The recent baby-boom/baby-bust cycle leads to welfare losses of about
5% of lifetime consumption for some cohorts. We argue that it is crucial for these re-
sultstomodelcorrectly thelabormarketdistortionsarisingfromthepensionsystem.
JEL classification: E62, H3, H55
Keywords: social security; pension design; optimal fiscal policy; demographic uncer-
tainty
∗We thank Alan Auerbach, Axel B¨orsch-Supan, Juan Carlos Conesa, Peter Diamond, Martin Gervais,
Ashok Kaul, Dirk Kru¨ger, Alice Schoonbroodt and seminar participants at SCE 2006 in Cyprus, Netspar
2007 Workshop in Tilburg, MIT, Institute of Advanced Studies in Vienna, Universitat Pompeu Fabra,
University of Frankfurt, University of Mannheim, LMU Munich and University of Wu¨rzburg for helpful
comments and discussions. Alex Ludwig gratefully acknowledges financial support from the German
NationalResearchFoundation(DFG)throughSFB504,theStateofBaden-Wu¨rttembergandtheGerman
Insurers Association (GDV).
†Mannheim Research Institute for the Economics of Aging (MEA); Universita¨t Mannheim; L13, 17;
68131 Mannheim; Germany; Email: [email protected].
‡Institute for Advanced Studies (IHS); Department of Economics & Finance; Stumpergasse 56; 1060
Vienna; Austria; Email: [email protected].
1
1 Introduction
All over the world, demographic transition processes are characterized by falling fertility
rates and increasing life-expectancy which leads to sharp decreases of working-age popula-
tion ratios and corresponding increases in old-age dependency ratios. This will put a strain
on public budgets, especially on existing public pay-as-you-go (PAYG) pension systems,
as is highlighted by the ongoing reform processes in many countries.
From an analytical perspective, it is useful to separate two distinct aspects of demo-
graphic change. First, the continuous increase in life expectancy will lead to a continuous
increase in the old-age dependency ratio (OADR) – the ratio of the population aged 65 and
older to the working agepopulationaged20 to 64– which will make current PAYG systems
unsustainable unless theretirement ageissignificantly increased. There isa largeliterature
dealing with this problem (cf. Kru¨ger and Ludwig (2007) and references there). A second
issue are the fluctuations in the fertility rate, notably the recent baby-boom/baby-bust
cycle, which lead to significant fluctuations in the OADR. In this paper we focus on the
second aspect of demographic change, and we therefore abstract from secular changes in
longevity.
The fluctuations of the OADR caused by changing fertility patterns affect different co-
horts in an unequal way, through their effects on factor prices and on the contributions and
benefits of pension systems. From an ex-ante perspective, these fluctuations are stochas-
tic, and the question arises as of how to design public pension systems such that they
contribute to an efficient sharing of the risks that arise through demographic fluctuations
(Bohn 2001). It is well known that a decentralized OLG economy cannot allocate all the
risks efficiently, since living generations cannot write contracts with generations not yet
born (Gordon and Varian 1988). Fiscal policy, in particular a public pension system, may
then have a role in making the intergenerational allocation more efficient. This is the
subject of this paper.
We address the issue of optimal pension policy in a standard overlapping generations
model with a realistic demographic structure. We compare three different types of policy:
(i) the first-best efficient response to demographic shocks, (ii) the second best optimal
policy, defined as the optimal Ramsey policy under commitment where the setup is similar
to Erosa and Gervais (2002) and (iii) several simple rules that give a stylized description of
pension systems currently in place in developed countries. Since the German government
has recently enacted a pension reform that explicitly reacts to changes in demographic
variables, our model is calibrated to German data, and we investigate the efficiency aspects
of the German pension system.
We place special emphasis on the distortionary effects of real-world pension systems on
household labor supply. These distortions are often exaggerated in the literature by the
assumption that pension contributions are tied to labor income, whereas benefits are lump
sum. We model the German pension system correctly by assuming that, within a cohort,
pension benefits are proportional to life-time contributions. According to this setup, the
pension system is distortionary only to the extent that the rate of return in the pension
system is lower than the market return on financial assets, such that households would
2
prefer not to contribute to the public system, but are forced to do so. The distortionary
effect of the pension system is aggravated by the fact that it adds to the already existing
labor supply distortions of general income taxation. Since the marginal excess burden is
rising in the tax level, the undesirable side effects of public pensions are the more severe
the higher is the “background distortion” from the general tax system. We highlight the
importance of this form of background distortion in this paper. We also show that a
realistic calibration of the labor supply elasticity is very important.
Our paper is most closely related to Bohn (2001) who analyzes the risk sharing prop-
erties of alternative pension policies and characterizes what an efficient response would be.
A main finding is that favourable factor price movements make the members of small co-
horts, the baby bust generation, better off, even after accounting for the additional burden
that these cohorts face by having to finance the pensions of the baby boom generation.
Contrary to conventional wisdom, an efficient response would require to even raise the
benefit levels of the baby boom generations. This is confirmed by Sanchez-Marcos and
Martin (2006) in a more realistic OLG setting. We show that this conclusion is reversed
once the distortionary nature of the public pension system is appropriately accounted for.
We evaluate simple policy rules by their distance from second-best outcomes in policy
space and also by the entailed welfare loss, in terms of a standard Utilitarian welfare
criterion. We find that a constant benefit rule is relatively close to the second-best policy
in most of our parameterizations. Moreoever, realistic simple rules outperform the Ramsey
solution if a different, but plausible criterion is used, namely to minimize the variability
of lifetime welfare. This suggests that, in the political process, other criteria than ex-ante
efficiency are dominant.
In our setup, the existence of a pension system reflects the preferences of a social
planner who wants to redistribute resources from future to current generations. Therefore,
our results are consistent with Krueger and Kubler (2006), who compare economies with
and without social security systems in the presence of aggregate technology shocks. They
find that only the early generations benefit from the introduction of a PAYG system. For
future generations, the crowding-out of private capital formation due to PAYG systems by
far outweighs the favorable effects of risk-sharing.1 For this paper, we accept the social
preferences that lead to a systematic redistribution from future to current generations.
We identify the respective weights of different cohorts in the social welfare function which
rationalize the actual size of the PAYG systems. Given those preferences, we ask how the
parameters of the pension system should be modified optimally in response to aggregate
shocks.
In a stylized two-period OLG model with stochastic production, Gottardi and Kubler
(2008) characterize the conditions for a social security system to provide an ex-ante Pareto
improvement. In their framework, the government can provide small stochastic lump sum
transfers from the young to the old, which are required to be non-negative. We differ
from their contribution by focusing on the distortionary aspects of pension systems, and
1Similar results have been reported for models featuring only idiosyncratic shocks by Conesa and
Krueger (1999).
3
by searching for optimal pension systems.
The outline of the paper is as follows. Section 2 presents the main elements of our
model. In Section 3 we discuss the properties of first best and second best allocations.
We also investigate the equivalence between a PAYG pension system and a system with
government debt. In Section 4 we then extend the model presented in Section 2 to a fully
specified model which can be used to analyze the quantitative questions we are interested
in. Section 5 presents our main results. Section 6 concludes.
2 The OLG Economy
2.1 Time and Risk
Time is discrete and extends from t = 0,...,∞. The only source of aggregate uncertainty
is the stochastic fertility rate. To keep the analysis focussed, we abstract from all other
potential sources of aggregate risk.
2.2 Demographic Processes
The exogenous driving force in our model is a time and age specific stochastic fertility
process. For simplicity, we take age-specific mortality rates as constant over time and we
abstract from migration. Starting from an initial age distribution of population in year 0,
the demographic distribution in each year t evolves according to
N ς for i > 0
t−1,i−1 i−1
N = (1)
t,i ( Ii=0ft−1,iNt−1,i for i = 0
P
where ς denotes age-specific mortality rates and f denotes age and time specific fertility
i t,i
rates.
2.3 The Production Sector
Firms are assumed to operate in a perfectly competitive market, using the constant returns
to scale production function
Y = F(K ,Z L ) (2)
t t−1 t t
where Y is output, K denotes the capital stock at the beginning of period t, and L is
t t−1 t
aggregate effective labor input in period t. Z is the technology level in period t that grows
t
at the constant rate g, hence Z = (1+g)tZ .
t 0
Profit maximization of the representative firm implies that the aggregate wage rate, w ,
t
and the rate of return to capital, r , are given by
t
w = F (K ,Z L ) (3)
t L t−1 t t
r = F (K ,Z L )−δ (4)
t K t−1 t t
4
where δ is the depreciation rate, and F and F denote the partial derivative of the
K L
production function with respect to the first and second argument, respectively.
2.4 The Household
The preferences of a household born in period t over distributions of consumption, C ,
t,i
and leisure, 1−L , are represented by the expected life-time utility
t,i
I
U = E βiπ U (C ,L ), (5)
t t i t+i,i t+i,i
i=iA
X
whereβ isthepuretimediscount factoroftheindividual householdandwhereexpectations
aretakenover theshocks tofertilityrates. Inadditiontopuretimediscounting, households
discount future utility with the probability of surviving to age i, which we denote by π .
i
Notice that π = i−1 ς , where ς is the conditional probability to survive from j to j+1.
i j=0 j j
Inthispaperweanalyzetheroleofthepensionsystemforintergenerationalrisk-sharing,
Q
not its merit in providing annuity markets, although this may be an important aspect in
reality. We therefore assume perfect annuity markets within cohorts (Yaari 1965). By
this idealized setup individual households are insured against the idiosyncratic mortality
risk. Accordingly, we define the adjusted net interest rate, rn , including annuitization and
t,i
capital taxation at rate τc as
t
1+r (1−τc)
1+rn = t t (6)
t,i ς
i−1
Households, indexed by i, receive labor income during their working period and pen-
sion income p when they are retired. Government taxes labor income at rates τl + τp,
t,i t
whereby τp are contributions to the pension system and τl is used to finance a lump sum
t
redistribution within the cohort, T . Net labor income is accordingly given by
t,i
w (1−τl −τp)L , (7)
t,i t t,i
where w = (cid:15) w is the age-specific gross wage, and (cid:15) is age-specific productivity, which
t,i i t i
displays a hump-shaped profile. Notice that (cid:15) = 0 for i ≥ iR, where iR is the retirement
i
age.
Collecting these components and denoting the financial wealth at the beginning of
period t by A , maximization of the household’s inter-temporal utility is subject to a
t−1,i−1
dynamic budget constraint given by
A +C = (1+rn )A +w (1−τl −τp)L +p +T . (8)
t,i t,i t,i t−1,i−1 t,i t t,i t,i t,i
Households start adult life with A = 0, because we abstract from intended or
t−1,iA−1
accidental bequests. The government lump-sum transfer T , and pension income p will
t,i t,i
be described in the next section.
5
2.5 The Government Sector
The government has access to a set of fiscal policy instruments and a commitment tech-
nology to implement its fiscal policy. It uses these instruments to implement efficient risk
sharing between generations. Pension payments, p , are financed by payroll taxes on labor
t,i
income, τp. In addition, the government has access to government debt, D , and capital
t t
income taxes, τc. Denoting by D the amount of outstanding government debt issued as
t t−1
bonds in period t−1, the government budget constraint is accordingly given by
I
D = (1+r )D + (p −τpw L −τcr A )N for all t. (9)
t t t−1 t,i t t,i t,i t t t−1,i−1 t,i
i=iA
X
Recall that w = 0 for i ≥ iR and p = 0 for i < iR. We describe the details of the
t,i t,i
pension system linking payments p to contributions τp in Sections 3.3 and 4.2.
t,i t,i
Exogenous to the government is the “background distortion” in the labor market, cap-
tured by the labor tax rate τl. We assume it is constant and finances a within-cohort
transfer T :
t,i
T = τlw L , i = iA,...,iR −1. (10)
t,i t,i t,i
The presence of this transfer system allows us to calibrate the degree of labor market
distortions in the economy. Background distortions are a metaphor for the effects of re-
distribution within the general tax system, which reflects preferences of society that we
do not model explicitly. They should also capture the effects of distortionary taxation to
finance public consumption. For the purpose of studying pension policy, we assume the
government cannot change τl.
2.6 Market Structure
Generations born in t can obviously not write an insurance contract with generations
born in t + τ, τ > 0. This is the fundamental form of market incompleteness in OLG
models that was pointed out by Gordon and Varian (1988). We assume additional market
incompleteness by focusing on an economy with only one asset, the physical capital stock
K , which was also assumed, e.g., by Krueger and Kubler (2006). Markets in this economy
t
are spot markets for consumption, labor, capital, all of which are assumed to be perfectly
competitive. We assume the economy to be closed.
2.7 Market Clearing and Aggregation
The aggregate resource constraint of the economy in period t is
Y = F(K ,Z L ) (11a)
t t−1 t t
= K −(1−δ )K +C (11b)
t t t−1 t
6
where
I I I
C = C N , L = (cid:15) L N , K = A N −D . (12)
t t,i t,i t i t,i t,i t t,i t,i t
i=0 i=0 i=0
X X X
3 First and Second-Best Allocations
3.1 Efficient Allocations
Two concepts of Pareto efficiency have been defined in the literature on OLG economies
(Demange 2002). The first is the concept of conditional or interim Pareto optimality,
which evaluates allocations conditional on the time period of birth of an agent. The
second concept is the concept of efficiency in the stronger sense of ex-ante efficiency which
we adopt here:
Definition 1. Let E U be the time 0 expectation of the life-time discounted utility of an
0 t
agent. A feasible allocation, x, is said to be ex-ante optimal if there is no other feasible
allocation, x˜, such that E U (x˜) ≥ E U (x) for all t with the inequality being strict for at
0 t 0 t
least one t.
This strong notion of efficiency requires that all the insurance possibilities between gen-
erations that exist at t = 0 are exploited. As Gordon and Varian (1988) have pointed out,
a decentralized equilibrium can in general not achieve ex-ante efficiency, because insurance
contracts with generations that are not yet born cannot be written.
Following Bohn (2001), we consider ex-ante efficient symmetric allocations.2 As in
Samuelson (1968), Atkinson and Sandmo (1980) and Erosa and Gervais (2002), an ex-ante
efficient allocation maximizes the Utilitarian social welfare function
∞
E Ω N U (13)
0 t t,0 t
t=−I
X
subject to the aggregate resource constraints (11) and (12) and the specifications of the
exogenous stochastic process for fertility.3 Here, Ω is the welfare weight of generation t
t
2Obviously, not all ex-ante efficient allocations solve (13), only those allocations that are symmetric
in the sense that they assign the same consumption and leisure to each born member of a generation.
From our ex-ante perspective, it is probably best to reinterpret the model of stochastic (but exogenous)
population as a model with deterministic population in the following way. Say that in each period t there
is a fixed set of households, some of which are randomly selected to be born and some of which are not.
In the latter case, households receive utility zero and are not affected by economic allocations. They do
therefore not appear in (13). Formula (13) gives a fixed weight to each born household. The concept of
Pareto-optimality is more problematic in the case of endogenous fertility, cf. Golosov et al. (2007).
3Social welfare functions of the above type with discounting of future generations have more recently
alsobeenadoptedindynasticmodelsbyBernheim(1989),CaplinandLeahy(2004)andFarhiandWerning
(2006,2008). See Stern(2006)fora criticalreviewofthe discussionregardingthe ethical justifications for
discounting in an intergenerationalcontext.
7
where Ω = 1. These weights reflect the relative preferences of a social planner for present
0
and future generations. Different weights give different efficient allocations.
The next proposition characterizes optimal allocations:
Proposition 1. An allocation is a solution to the optimization problem (13) if and only
if it satisfies (11), (12) and the following first order conditions:
−U (C ,L ) = Z (cid:15) F (K ,Z L )U (C ,L ) (14a)
L t,i t,i t i L t−1 t t C t,i t,i
U (C ,L ) = βE (1+F (K ,Z L )−δ)U (C ,L ) (14b)
C t,i t,i t K t t+1 t+1 C t+1,i+1 t+1,i+1
U (C ,L ) Ω
C t,i t,i = t−jβj−i, ∀i,j ∈ (0,...,I) (14c)
U (C ,L ) Ω
C t,j t,j t−i
Proof. See Appendix A.3.
Conditions (14a) and (14b) are the first order conditions of the household problem in
an undistorted decentralized equilibrium with perfect annuity markets. Condition (14c)
illustrates the perfect risk-sharing between cohorts. The ratio of marginal utilities from
consumption of different generations is constant. This means that the effects of a shock
(past or present) are spread equally across generations in t. In a decentralized equilibrium,
there is no mechanism to achieve this. However, a government can implement such an
allocation if it has stochastic lump sum taxes at its disposal.
3.2 The Risk-Free Economy on the Balanced Growth Path
We next concentrate on the implications of our model along the balanced growth path
of a risk-free economy, in which variables grow at constant rates. We call such a path
a deterministic steady state. Along the balanced growth path, labor input is constant,
capital and output grow at rates g+n, where n is the steady state population growth rate.
The marginal product of capital is constant at 1+r¯ = 1+F (K (1+g +n)t,L ,1)−δ
K 0 0
and the wage rate grows at the constant rate g.
To get balanced growth, we make the following homogeneity assumptions:
Assumption 1. The marginal utility of consumption, U (C ,L ), is homogeneous of
C t,i t,i
degree −θ in C .
t,i
Assumption 2. Social weights are Ω = ωt.
t
With these assumptions we can derive the following relationship to hold between the
exogenous growth rate of technology, g, the steady state interest rate, r¯, and the welfare
weights, ω:
Proposition 2. Consider the steady state of a risk-free economy that maximizes (13) and
denote the steady state interest rate by r¯. Then ω = (1+g)θ .
1+r¯
(cid:16) (cid:17)
Proof. See Appendix A.3.
8
Therelationshipbetween thegovernment discount factorandtheinterest rateinPropo-
sition 2 is also referred to as the modified golden rule property (Samuelson 1968) and has
some interesting implications. First, it shows that a competitive deterministic economy
without government, and an efficient allocation that maximizes (13), have the same steady
state if and only if
(1+g)θ
ω = (15)
n
1+r¯
d
where r¯ is the interest rate of the competitive economy. This means that a government
d
with a discount rate given in Equ. (15) is inactive in steady state; it does not interfere with
the private allocation. We call such a government “neutral”:
Definition 2. A government with a discount factor ω satisfying (15) is called a neutral
n
government. A government with a discount factor ω < ω is called an impatient govern-
n
ment.
An impatient government gives relatively higher weight to currently living generations
and will use its policy instruments to redistribute from future to current generations. We
do not consider the case of a “patient” government with ω > ω .
n
More generally, in an allocation maximizing (13), the interest rate is given by
(1+g)θ
r¯= −1. (16)
ω
Changes of any parameter other than g, θ and ω have no effect on the optimal interest rate.
For example, if we change household preferences in a way that generates higher savings,
the government sets policy instruments such that it crowds out private savings until the
condition (16) is met.
3.3 The Fiscal Policy Setup: Debt vs. Pensions
It is well known that a pension system and government debt are basically equivalent policy
instruments, at least in deterministic models or models with complete markets. Here we
derive conditions under which equivalence holds in a stochastic incomplete markets model
with distortive taxation. This means that all taxes and transfers that a household has to
pay are linearly related to the household decisions on labor supply and saving. There are
no lump sum taxes or transfers. For reasons of space, we confine ourselves to the case of
two-periodlives, where households work inthe first periodandretire inthe second. Similar
results hold with many cohorts, if the government has age-dependent taxes available; this
is investigated in detail in Ludwig and Reiter (2008).
In a stochastic setup, the pension system serves two purposes. First, it shifts resources
from future to current generations, which we assume reflects the preferences of the policy
maker. Second, it helps to spread the effects of shocks efficiently between different gen-
erations. To highlight both aspects, we distinguish between three different fiscal policy
regimes:
9
1. Debt policy (DP). The instruments are government debt, a labor income tax at rate
τd and the capital tax τc. The capital tax is state dependent, i.e., the tax rate that
t t
applies in period t is only determined in t. Government debt follows
(1+r )d N −d N = τdw L N +τcr (k +d )N for all t.
t t−1,0 t−1,0 t,0 t,0 t t,0 t,0 t,0 t t t−1,0 t−1,0 t−1,0
(17)
Here, k and d denote asset holdings in the form of capital and debt, respectively,
t,0 t,0
such that k + d = A . As only the young are savers in our simplified two
t,0 t,0 t,0
generations economy, we have that K = k N and D = d N . We assume a
t t,0 t,0 t t,0 t,0
no-Ponzi condition on the government.
The household budget constraints are
k +d +C = w (1−τd −τl)L +T (18a)
t,0 t,0 t,0 t,0 t t,0 t,0
1+r (1−τc )
C = t+1 t+1 (k +d ) (18b)
t+1,1 t,0 t,0
ς
t,0
and the household first order conditions are
1+r (1−τc )
U (C ,L ) = βς E t+1 t+1 U (C ,0) (19a)
C t,0 t,0 t,0 t C t+1,1
ς
(cid:20) t,0 (cid:21)
−U (C ,L ) = w (1−τd −τl)U (C ,L ) (19b)
L t,0 t,0 t,0 t C t,0 t,0
2. Pension policy with a predetermined pension factor (PP1). The instruments are
pension contributions τp, the capital tax τc, and a pension benefit factor ˜b . The
t t t
capital tax is specified as in DP. Pension income of the old in period t+1 is given
by
1+r (1−τc )
p = τp w L t+1 t+1 ˜b . (20)
t+1,1 t,0 t,0 t,0 ς t
t,0
It is the product of three components: the past contributions to the pension system,
τp w L ; the adjusted interest on contributions, 1+rt+1(1−τtc+1); the pension factor,
t,0 t,0 t,0 ςt,0
˜
b , which is already fixed in period t (policy instruments carry a tilde if they are
t
predetermined, which means that the rate applied in t+1 is already determined in
t). The government budget is balanced in every period:
1+r (1−τc)
˜b t t τp w L N = τp w L N +τcr k N for all t.
t−1 ς t−1,0 t−1,0 t−1,0 t,1 t,0 t,0 t,0 t,0 t t t−1,0 t−1,0
t−1,0
(21)
The household budget constraints are
k +C = w (1−τp −τl)L +T (22a)
t,0 t,0 t,0 t t,0 t,0
1+r (1−τc )
C = t+1 t+1 k +˜b τpw L (22b)
t+1,1 ς t,0 t t t,0 t,0
t,0
h i
10