## Solow Growth Model: What If _s_ is the Net Rather than the Gross Savings Rate? ¶

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) $Y^{*net}_t = \left( \frac{n + g}{n+g+\delta s} \right) \left( \frac{s}{n+g + \delta s} \right)^{(α/(1-α))}L_tE_t$