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 $

 


 

Solow Growth Model: What If $ s $ is the Net Rather than the Gross Savings Rate?

Catch Our Breath—Further Notes: