Proceedings from the ECFIN Workshop "The budgetary implications of structural reforms" - Brussels, 2 December 2005

Appendix A: The welfare state and the social
contract - PAYG-financing and sustainable tax
rates

A simplified version of the social contract can be represented in a standard
two-period overlapping generations model with generations in the work force
(the young) overlapping with retirees (the old). N_t is the number of persons in
the generation being young in period t (and therefore old in period t +1), a1t
denotes per capita services received from the public sector as young in period
t, and a2t per capita services received by the old in period t. Pensions paid to
old in period t denoted pt . The total income of those being active in period t is
denoted y_1t, and it is taxed by a proportional tax rate τ .

The primary balance of the public sector in period t is

bt = Ntτty1t - Nta1t - Nt-1(pt + a2t)

= Nt [τty1t - a1t - (1 + ρ_t)(pt + a2t)]

where 1+ρ is the demographic dependency ratio³⁰ in period t, i.e. 1+ρ_t.It
follows that ρ is negative when the population is growing, and positive when
it is decreasing. An increase in longevity prolongs the "retirement period" and
increases the demographic dependency ratio and can thus be interpreted as a
reason why ρ is positive.

In net terms the exchange of the young generation with the public sector is

Γ1t = Nta1t - Ntτ ty1t

and for the old it is

Γ2t = Nt-1(pt + a2t)

Given policy rules for welfare service, pensions and taxes - p(), a1 (),a2() and
τ () - can be realized if and only if

PV(b)+D0 ≥0                        (7)

where D0 is the initial public debt.

In the following we consider policy rules where

pt  =  γyt                                 (8)

a1_t   =  α1yt                                  (9)

a2t  =  α2yt                              (10)

For pensions and welfare services it follows that they are proportional to cur-
rent income. The first part of this assumption implies a fixed relation between

³⁰ The fact that not all working people survive and reach the pensioner stage can easily be
included. Denote by Ot the number of pensioners in period t out of generation Nt born in
t - 1, where Ot =(1- φ_t-1 )Nt-1 , and φ is the likelihood of dying young. The demographic
dependency ratio is now Nt = 1+e ≡ (1-Φt-ι)(1+pt) < (1+pt). Beyond this reinterpretation
of the change in the demographic dependency ratio, the analysis remains unchanged.

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