Last edited: 2019-08-17
Definition of capital-output ratio:
(1) $ κ_t = \frac{K_t}{Y_t} $
Factor accumulation:
(2) $ \frac{dL_t}{dt} = nL_t $
(3) $ \frac{dE_t}{dt} = gE_t $
(4) $ \frac{dK_t}{dt} = s(Y_t - \delta K_t) $
(5) $ \frac{dK_t}{dt} = s(1 - \delta κ_t)Y_t $
Production function:
(6) $ Y_t = K_t^α(L_tE_t)^{(1-α)} $
(7) $ Y_t = κ_t^{(α/(1-α))}L_tE_t $
(8) $ κ^* = \frac{s(1 - \delta κ^*)}{n+g} $
(9) $ κ^* + \delta κ^* \frac{s}{n+g} = \frac{s}{n+g} $
(10) $ κ^* \left( \frac{n + g + \delta s}{n+g} \right) = \frac{s}{n+g} $
(11) $ κ^* = \frac{s}{n+g + \delta s} $
(12) $ Y^*_t = \left( \frac{s}{n+g + \delta s} \right)^{(α/(1-α))}L_tE_t $
(13) $ \ln(Y^*_t) = (α/(1-α)) \left( \ln(s) - \ln(n+g+\delta s) \right) + \ln(E_t) + \ln(L_t) $
(13) $ \ln(Y^{*net}_t) = (α/(1-α)) \left( \ln(s) - \ln(n+g+\delta s) \right) + \ln(E_t) + \ln(L_t) + \ln(n+g) - \ln(n+g+\delta s)) $
(14) $ Y^{*net}_t = \left( \frac{n + g}{n+g+\delta s} \right) \left( \frac{s}{n+g + \delta s} \right)^{(α/(1-α))}L_tE_t $