Munich Personal RePEc Archive
Agglomeration economies and optimal efficiency wage
Jellal, Mohamed
Al Makrîzi Instittut D’économie, Rabat , Morocco
8 August 2014
Online at https://mpra.ub.uni-muenchen.de/57836/
MPRA Paper No. 57836, posted 10 Aug 2014 10:02 UTC
).
𝐹 𝑛𝑒 𝑤 𝑒 𝑤 𝑤
𝑛
𝑒′ 𝑤 > 0 𝑒" 𝑤 < 0
𝐹 𝑛,𝑒 𝑤
𝐴(𝑁)𝐹 𝑛,𝑒 𝑤 𝐴(𝑁)
𝑁 = 𝑛 𝑖 𝑑𝑖 𝐴′ 𝑁 > 0 𝐴′′ 𝑁 < 0
𝐴 𝑁
max
𝑙,𝑤Π 𝑛 , 𝑤 = 𝐴 𝑁 𝐹 𝑛 , 𝑒 𝑤 − 𝑤𝑛
𝑛 = 𝑁 = 𝑛 𝑖 𝑑𝑖
𝑛 𝑤
𝜕Π(n,w)
𝜕𝑛
= 𝐴 𝑁 𝐹
𝑛𝑛 , 𝑒 𝑤 − 𝑤 = 0
𝜕Π(n,w)
𝜕𝑤
= 𝐴 𝑁 𝐹
𝑒𝑛 , 𝑒 𝑤 𝑒
′( 𝑤 ) − 𝑛 = 0 𝐹
𝑛=
𝜕𝐹𝜕𝑛
𝐹
𝑒=
𝜕𝐹𝜕𝑒
𝜂
𝑛=
𝑛𝐹
𝜕𝐹
𝜕𝑛
𝜂
𝑒=
𝑒𝐹
𝜕𝐹
𝜕𝑒
𝜂
𝑎=
𝑁𝐴 𝑑𝐴 𝑑𝑁
𝜀 𝑤 =
𝑤𝑒
𝑒′(𝑤) 𝑒(𝑤)
𝜀 𝑤 =
𝑤𝑒
𝑒′(𝑤) 𝑒(𝑤)
=
𝜂𝑛𝜂𝑒
𝑛𝐴 𝑁 𝐹𝑛 𝑛,𝑒 𝑤 = 𝑤𝑛
𝐹𝑒 𝑛,𝑒 𝑤 𝑒′ 𝑤 𝑤 = 𝑛𝐹𝑛 𝑛,𝑒 𝑤
𝑒 𝑤 𝐹𝑒 𝑛,𝑒 𝑤
𝐹 𝑛,𝑒 𝑤 𝑒′(𝑤)
𝑒(𝑤) 𝑤 =𝑛𝐹𝑛 𝑛,𝑒 𝑤 𝐹(𝑛,𝑒(𝑤)
𝜂
𝑛𝜂
𝑒𝜀 𝑤 =
𝑤𝑒
𝑒′(𝑤) 𝑒(𝑤)
=
𝜂𝑛𝜂𝑒
𝜀 𝑤 ≶ 1 ⟺ 𝜂
𝑛≶ 𝜂
𝑒𝐴 𝑁 𝐹 𝑛,𝑒 𝑤
𝜂
𝑛𝜂
𝑒𝜂
𝑛𝜂
𝑒max
𝑙,𝑤Π 𝑛 , 𝑤 = 𝐴 𝑁 𝐹 𝑛 , 𝑒 𝑤 − 𝑤𝑛 𝑛 = 𝑁 = 𝑛 𝑖 𝑑𝑖
𝜕Π(n,w)
𝜕𝑛
= 𝐴
′𝑁 𝐹 𝑁 , 𝑒 𝑤 + 𝐴 𝑁 𝐹
𝑛𝑁 , 𝑒 𝑤 − 𝑤 = 0
𝜕Π(n,w)
𝜕𝑤
= 𝐴 𝑁 𝐹
𝑒𝑁 , 𝑒 𝑤 𝑒
′( 𝑤 ) − 𝑁 = 0
𝜀 𝑤
∗=
𝜂𝑛𝜂𝑒
+
𝜂𝑎𝜂𝑒
𝑁𝐴′ 𝑁 𝐹 𝑁,𝑒 𝑤 +𝑁𝐴 𝑁 𝐹𝑛 𝑛,𝑒 𝑤 = 𝑤𝑁
𝑁𝐴′ 𝑁 𝐹 𝑁,𝑒 𝑤 +𝑁𝐴 𝑁 𝐹𝑛 𝑁,𝑒 𝑤 = 𝐴 𝑁 𝐹𝑒 𝑁,𝑒 𝑤 𝑒′(𝑤)𝑤
𝜂
𝑛=
𝑛𝐹
𝜕𝐹
𝜕𝑛
𝜂
𝑒=
𝑒𝐹
𝜕𝐹
𝜕𝑒
𝜂
𝑎=
𝑁𝐴 𝑑𝐴 𝑑𝑁
𝑁𝐴 𝑁 𝐹𝑛 𝑁,𝑒 𝑤 1 +𝜂𝑎
𝜂𝑒 =𝐴 𝑁 𝑒(𝑤)𝐹𝑒 𝑁,𝑒 𝑤 𝑒′𝑒((𝑤𝑤))𝑤
𝜀 𝑤
∗=
𝜂𝑛𝜂𝑒
+
𝜂𝑎𝜂𝑒
𝜀 𝑤
∗= 𝜀 𝑤 + 𝜂
𝑎𝜂
𝑒𝜀 𝑤
∗≶ 1 ⟺ 𝜂
𝑛+ 𝜂
𝑎≶ 𝜂
𝑒𝐴 𝑁 𝐹 𝑛,𝑒 𝑤
𝜂
𝑎max
𝑙,𝑤Π 𝑛 , 𝑤 = 𝐴 𝑁 𝐹 𝑛 , 𝑒 𝑤 − 𝑤𝑛 + 𝑠𝑛 − 𝑇 𝑛 = 𝑁 = 𝑛 𝑖 𝑑𝑖 𝑇 = 𝑠𝑛 𝑖 𝑑𝑖
𝑠 𝑇
𝜀 𝑤 =
𝜂𝑛𝜂𝑒 𝑤 𝑤−𝑠
𝑠
∗= 𝜂
𝑎𝜂
𝑛+ 𝜂
𝑎𝑤
∗𝜀 𝑤
∗= 𝜂
𝑛𝜂
𝑒+ 𝜂
𝑎𝜂
𝑒𝜀 𝑤 = 𝜂𝑛 𝜂𝑒
𝑤 𝑤 − 𝑠
𝑤
∗= 𝑤
𝜂𝑛
𝜂𝑒 + 𝜂𝑎 𝜂𝑒 = 𝜂𝑛
𝜂𝑒
𝑤 𝑤 − 𝑠
𝑠 = 𝑠
∗=
𝜂𝑎𝜂𝑛+𝜂𝑎