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Econ 102 Fall 2014 The mathematical derivation of the simple Solow Model: (You have already received the lecture notes on Solow model and detailed explanations in the previously sent email. The following only exposes some of the step in the mathematical derivation.) Production Function: Yt K t L1t (1) Law of motion for capital stock: K t 1 K t K t I t K t (1 ) sYt (2) Law of motion for Labor: Lt L0 (1 n) t (3) The same model can be defined in terms of per capita variables : yt K Yt and k t t Lt Lt Using equation (1): Yt K t L1t K t yt kt Lt Lt Lt (4) Using equation (3): K t 1 K t (1 ) sYt kt (1 ) syt kt (1 ) sk t Lt Lt (5) Furthermore K t 1 K t 1 (1 n) K t 1 (1 n) kt 1 (1 n) Lt Lt (1 n) Lt 1 Hence equation (5) becomes kt 1 (1 n) kt (1 ) sk t (6) Which is the new law of motion for capital per capita. 1 In a steady state equilibrium the capital per capita will not be changing and will be constant over time. Therefore: k t 1 k t k SS Substitute this condition into equation (6) k SS (1 n) k SS (1 ) s(k SS ) Simplify the above equation with a very simple step by step derivation: k SS (1 n) k SS (1 ) s(k SS ) k SS (1 n 1 ) s(k SS ) k SS (n ) s(k SS ) k SS s (k SS ) (n ) k SS s SS (k ) (n ) (k SS ) (1 ) (k s (n ) 1 SS (1 ) 1 ) s (n ) 1 1 The above condition gives us the steady state condition for capital per capita: s k SS (n ) 1 1 The steady state condition for the output per capita will then be: s y (k SS ) (n ) 1 2