use St = and St=It : s∙Yt Kt+1 = - Kt s∙Yt divide by K to express it as the growth rate in K: = - Kt+1/Kt  s∙Yt/Kt - Is k=K/N growing, shrinking or staying the same? Growing if: > (= n) Kt+1/Kt t+1/Nt - > 0 Kt+1/Kt t+1/Nt - n >0 - s∙Yt/Kt multiply last condition through by Kt/Nt: - ( >0 s∙Yt/Nt n)∙Kt/Nt or (using definitions of y and k): - ( >0 s∙yt n)∙kt - So: K/N is growing if: - ( >0 s∙yt n)∙kt K/N is shrinking - ( <0 if: s∙yt n)∙kt K/N is constant - ( =0 if: s∙yt n)∙kt - Look more closely at these conditions: syt = amount of savings and investment (expressed per N) i.e. amount of new capital being created (per N). ( = amount of new capital needed to hold K/N constant (expressed n)∙kt
use St = s∙Yt and St=It : Kt+1 = s∙Yt - Kt divide by K to express it as the growth rate in K: Kt+1/Kt = s∙Yt/Kt -  - Is k=K/N growing, shrinking or staying the same? Growing if: Kt+1/Kt > t+1/Nt (= n) Kt+1/Kt - t+1/Nt > 0 s∙Yt/Kt - - n >0 multiply last condition through by Kt/Nt: s∙Yt/Nt - ( n)∙Kt/Nt >0 or (using definitions of y and k): s∙yt - ( n)∙kt >0 - So: K/N is growing if: s∙yt - ( n)∙kt >0 K/N is shrinking if: s∙yt - ( n)∙kt <0 K/N is constant if: s∙yt - ( n)∙kt =0 - Look more closely at these conditions: syt = amount of savings and investment (expressed per N) i.e. amount of new capital being created (per N). ( n)∙kt = amount of new capital needed to hold K/N constant (expressed