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

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w a t+1 =(1+g) w a t . By setting τ t+1 = τ t = τ in eq. 10 we obtain the following value for the steady state social security contribution rate: τ = (1 + n)(1+g) w − (1 + r)wa 2(1 + n)(1+g) w − (1 + r) w a + (16) q (1 + n) 2 (1 + g) 2 w 2 − A [2(1 + n)(1+g) w − (1 + r) w a ] − 2(1 + n)(1+g) w − (1 + r) w a (17) Simple algebra shows that τ ∈ [0, 1/2) and that ∂τ/∂n < 0. Hence, aging increases the social security contribution rate at steady state and — by eq. 4— early retirement. Q.E.D. 5.3 Proof of proposition 4 Consider the steady state social security contribution rate at eq. 16. Simple algebra shows that ∂τ/∂g < 0. Hence, productivity slow-downs increase the social security contribution rate at steady state and — by eq. 4— early retirement. Q.E.D. 5.4 Proof of proposition 5 At time t, when the adults’ and young’s labor income drops, the median voter will not change the social security contribution rate from its steady state level, in order to validate the expectations of the previous median voter at time t − 1. She will however expect the tax rate to be changed at t + 1 due to this drop in labor income. By eq. 12, it is thus immediate to see that z t =1− τ − δwt a > z =1− τ − δwt−1, a sincewt a
∂τ t+1 ∂w = − δ (1 + n t+1 ) ϕ +1 + Θ s µ 1 − δ(wa t+1) 2 2 w t+1 2 − 2[A−(1+r)wa t (1−τ t−δw a t )2 ] (1+n t )w t+1 where Θ = Ã δ 1 − δ ¡ wt+1 a (1 + n t+1 ) ϕ +1 ¢ 2 ! w t+1 − [(1 + n t+1) ϕ +1] £ A − (1 + r)w a t z t 2 ¤ (1 + n t )(w t+1 ) 2 − (1 + r)z t(z t − 2δw a t ) (1 + n t )w t+1 Notice that we cannot sign ∂τ t+1 /∂w; however, it is easy to show that a sufficient µ condition for Θ > 0isthatA w t+1 . By eq. 12, we thus have that ∂z t+1 ∂w δ = (1 + n t+1 ) ϕ +1 − Θ 2(1+n t+1 )(w t+1 ) 2 − δ = (1 + n t+1 ) ϕ = −δ (1 + n t+1 ) ϕ +1 − Θ 2(1+n t+1 )(w t+1 ) 2 < 0 since Θ > 0. Q.E.D. 29
Page 1 and 2: The Evolution of Retirement by J. I
Page 3 and 4: The Evolution of Retirement ∗ J.
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 and 28: [33] Jimeno, J. F., 2002b, “Incen
Page 29: Finally, to make sure that the uppe
Page 33 and 34: Table 4. Number of pensioners INDEP
Page 35: RELACION DE DOCUMENTOS DE FEDEA DOC

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