The Evolution of Retirement by J. Ignacio Conde-Ruiz* Vincenzo ...

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5 Appendix 5.1 Proof of proposition 2 The first order condition of the median voter is: −z t w a t (1 + r)+ ∂p t+1 ∂τ t+1 Q 0 (z t ) Z 0 (τ t )=0 where ∂p t+1 =(1+n t )(1− 2τ t+1 ) w t+1 and Z 0 (τ t )=−1 ∂τ t+1 Thus, the first order condition becomes −z t w a t (1 + r) − (1 + n t ) w t+1 Q 0 (z t )+2(1+n t ) w t+1 Q 0 (z t ) Q (z t )=0 Integrating the above equation with respect to z t ,weobtain A − z 2 t w a t (1 + r) − 2(1+n t ) w t+1 Q +2(1+n t ) w t+1 Q 2 =0 where A is a constant of integration. Solving the equation w.r.t. Q, wehave s Q (τ t )= 1 2 ∓ 1 1 − 2A − 2(1 + r)wa t (1 − τ t ) 2 . 2 (1 + n t )w t+1 Since τ t+1 = Q (τ t ) represents a tax rate, it has to be that Q ∈ [0, 1]. Thus, we have that τ t+1 = Q (τ t )= 1 2 − 1 2 s 1 − 2A − 2(1 + r)wa t (1 − τ t ) 2 (1 + n t ) w t+1 and the following restrictions on the free parameter A: (1 + r)w a t (1 − τ t ) 2 ≤ A ≤ (1 + n t) w t+1 2 +(1+r)w a t (1 − τ t ) 2 . Notice that, since the lower constraint is maximized for τ t+1 = 0, while the upper constraint is minimized for for τ t+1 =1,asufficient condition for Q ∈ [0, 1] is A ∈ [(1 + r)wt a , (1 + n t) w t+1 ]. 2 26
Finally, to make sure that the upper bound is indeed larger that the lower bound, we need to assume that (1 + n t ) w t+1 > 2(1 + r)wt a ; or, analogously, that [(1 + n t−1 ) ϕ +1]/2 > (1 + r) / [(1 + n t )(1+g)]. Q.E.D. 5.2 Proof of proposition 3 Consider the economy at its steady state and recall that eq. 10 provides the law of motion of the social security contribution rate. At time t + 1, a decrease in the population growth rate, n t+1 τ t ,and — from eq. 4 — so does the early retirement, since z t+1 =1− τ t+1
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Page 5 and 6: In this paper, we aim at providing
Page 7 and 8: social security expenditure seems t
Page 9 and 10: similar result is obtained if we re
Page 11 and 12: t +1andr is the interest rate. To a
Page 13 and 14: economy, while in models that conce
Page 15 and 16: supported by a Markovian politico-e
Page 17 and 18: τ — the use of early retirement
Page 19 and 20: changes in income or wealth on the
Page 21 and 22: of equilibrium social security cont
Page 23 and 24: we emphaze the relevance of the inc
Page 25 and 26: [10] Costa, D. (1998). “The Evolu
Page 27: [33] Jimeno, J. F., 2002b, “Incen
Page 31 and 32: ∂τ t+1 ∂w = − δ (1 + n t+1
Page 33 and 34: Table 4. Number of pensioners INDEP
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