# Simple derivation

```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
```