Munich Personal RePEc Archive
Education private and social returns an optimal taxation policy
Jellal, Mohamed
Al Makrîzi Institut d’économie , Rabat Morocco
8 July 2014
Online at https://mpra.ub.uni-muenchen.de/57190/
MPRA Paper No. 57190, posted 09 Jul 2014 21:15 UTC
𝐻 = 𝐿 𝐿
𝛿 𝑓 𝑎 = 𝛿𝑒−𝛿𝑎
𝑠 (𝑠,𝑆)
𝑆
𝑠
Max𝑠 𝑠 (𝑤 𝑡 𝑠,𝑆 (1 −𝜏) + 𝑇(𝑡))𝑒− 𝛿+𝑟 𝑡𝑑𝑡 𝑇(𝑡) = 𝜏𝑤 𝑡 𝑆,𝑆
Ex post 𝑠 = 𝑆
𝑤(𝑡) 𝑔
𝜏 𝑇(𝑡)
1 − 𝜏 𝜕
(𝑠∗,𝑆)
(𝑠𝜕𝑠∗,𝑆) = 𝑟 − 𝑔 +𝛿
𝜖 =
𝜕(𝑠∗,𝑠∗)
𝜕𝑠
(𝑠∗,𝑠∗) 𝜂 =
𝜕(𝑠∗,𝑠∗)
𝜕𝑆
(𝑠∗,𝑠∗)
1 − 𝜏 𝜕 𝑠∗
,𝑆
𝑠𝜕𝑠∗,𝑆 = (1 − 𝜏) 𝜖
𝑠∗ = 𝑟 − 𝑔 +𝛿
𝑠∗ = 𝜖(1 − 𝜏) 𝑟 − 𝑔 +𝛿
∗ = 𝜖(1 − 𝜏) 𝑟 − 𝑔 +𝛿
𝜀+𝜂
𝑠,𝑆 = 𝑠𝜖.𝑆𝜂 𝑆𝜂
Max𝑠 𝑠 (𝑤 𝑡 𝑠,𝑆 (1 −𝜏) + 𝑇(𝑡))𝑒− 𝛿+𝑟 𝑡𝑑𝑡 𝑇(𝑡) = 𝜏𝑤 𝑡 𝑆,𝑆
Ex ante 𝑠 = 𝑆
Max𝑆 𝑤 𝑆 0 𝑆,𝑆 𝑒− 𝛿+𝑟 −𝑔 𝑡𝑑𝑡
𝑆∗ = 𝜖 +𝜂 𝑟 − 𝑔 +𝛿
𝜕(𝑆∗,𝑆∗)
𝜕𝑠
(𝑆∗,𝑆∗) +
𝜕(𝑆∗,𝑆∗)
𝜕𝑆
(𝑆∗,𝑆∗) = 𝜖+𝜂
𝑆∗ = 𝑟 − 𝑔 +𝛿
𝑆∗ = 𝜖 +𝜂 𝑟 − 𝑔 +𝛿
𝑆∗ = 𝑠∗ + 𝜂 +𝜖𝜏 𝑟 − 𝑔 +𝛿
𝑠∗ = 𝜖(1−𝜏)
𝑟 −𝑔 +𝛿 𝑆∗ =
𝜖+𝜂
𝑟 −𝑔 +𝛿 𝑆∗ − 𝑠∗ = 𝜂+𝜖𝜏
𝑟 −𝑔 +𝛿
𝐷 = 𝜂+𝜖𝜏
𝑟 −𝑔 +𝛿 𝜂
𝜏
𝐻∗ = 𝜖 + 𝜂 𝑟 − 𝑔 +𝛿
𝜀+𝜂
𝐻∗ = 𝑆∗,𝑆∗ = 𝑆∗𝜖+𝜂 𝑆∗ = 𝜖+𝜂
𝑟 −𝑔 +𝛿
𝜏∗ = −𝜂 𝜖
𝑆∗ = 𝑠∗ + 𝜂 +𝜖𝜏 𝑟 − 𝑔 +𝛿 𝑆∗ = 𝑠∗ 𝐷 = 𝜂+𝜖𝜏
𝑟 −𝑔 +𝛿 = 0
𝑟 − 𝑔 +𝛿) > 0 𝜂 +𝜖𝜏 = 0 𝜏 = 𝜏∗ =
−𝜂𝜖