simplified output-capital relationship  

• assumes that population and participation rate are constant, and there is no technological progress
  • constant population, participation rate >> constant employment N
  • no technological progress >> technology doesn't change over time >> no shifts in production function over time
• Y/N = f(K/N) = F(K/N, 1)
  • works for functions satisfying constant returns to scale
• investments - assumes no public saving
  • I = S + (T-G) = S = sY
  • I = sY, where s = saving rate

investment-capital accumulation  

• depreciation rate (d) - proportion of capital stock that becomes worthless each year
• K_t+1 = (1-d)K_t + I_t
  • K_t+1/N = (1-d)K_t/N + I_t/N
  • K_t+1/N = (1-d)K_t/N + (sY_t)/N
  • K_t+1/N - K_t/N = (sY_t)/N - dK_t/N
• K_t+1/N - K_t/N = D(K/N) = sf(K_t/N) - dK_t/N
  • change in capital equal to difference of investment and capital depreciation

steady state - in long run, investment equals capital depreciation  

• steady state
• depreciation (dK_t/N)
• investment (sf(K_t/N)
• output (f(K_t/N)
• in long run, capital per worker will always converge to the steady state
• start to the left, investment is greater than depreciation, so K/N will increase to steady state
• start to the right, depreciation is greater than investment, so K/N will decrease to steady state