Gross Saving and Growth in the RCK Model
In the neoclassical growth model with labor-augmenting technological
progress at rate γ \pGro γ u ( c ) = c 1 − ρ / ( 1 − ρ ) \util(c) = c^{1-\CRRA}/(1-\CRRA) u ( c ) = c 1 − ρ / ( 1 − ρ ) ϑ \DiscRate ϑ δ \delta δ γ \pGro γ
C ˙ C = ρ − 1 ( f ′ ( k ) − δ − ϑ ) = γ \begin{gathered}\begin{aligned}
\frac{\dot{C}}{C} & = \CRRA^{-1}(f^{\prime}(k)-\delta-\DiscRate) \\
& = \pGro
\end{aligned}\end{gathered} C C ˙ = ρ − 1 ( f ′ ( k ) − δ − ϑ ) = γ which implies that
f ′ ( k ) = ρ γ + ϑ + δ α k α − 1 = ρ γ + ϑ + δ k = [ ρ γ + ϑ + δ α ] 1 α − 1 . \begin{gathered}\begin{aligned}
f^{\prime}(k) & = \CRRA \pGro + \DiscRate + \delta \\
\alpha k^{\alpha-1} & = \CRRA \pGro + \DiscRate + \delta \\
k & = \left[\frac{\CRRA \pGro + \DiscRate + \delta}{\alpha}\right]^{\frac{1}{\alpha-1}}.
\end{aligned}\end{gathered} f ′ ( k ) α k α − 1 k = ρ γ + ϑ + δ = ρ γ + ϑ + δ = [ α ρ γ + ϑ + δ ] α − 1 1 . The aggregate gross saving rate is defined as
s = y − c y = k α − c k α = 1 − c / k α . \begin{gathered}\begin{aligned}
s & = \frac{y-c}{y} \\
& = \frac{k^{\alpha}-c}{k^{\alpha}} \\
& = 1-c/k^{\alpha}.
\end{aligned}\end{gathered} s = y y − c = k α k α − c = 1 − c / k α . In steady-state by definition
k ˙ = 0 \begin{gathered}\begin{aligned}
\dot{k} & = 0
\end{aligned}\end{gathered} k ˙ = 0 but from the capital accumulation equation we know that
k ˙ = k α − ( δ + γ ) k − c \begin{gathered}\begin{aligned}
\dot{k} & = k^{\alpha}-(\delta+\pGro)k - c
\end{aligned}\end{gathered} k ˙ = k α − ( δ + γ ) k − c so in steady-state
c = k α − ( δ + γ ) k . \begin{gathered}\begin{aligned}
c & = k^{\alpha}-(\delta+\pGro)k.
\end{aligned}\end{gathered} c = k α − ( δ + γ ) k . This can be substituted into (3)
s = 1 − k α − ( δ + γ ) k k α = ( δ + γ ) k 1 − α \begin{gathered}\begin{aligned}
s & = 1-\frac{k^{\alpha}-(\delta+\pGro)k}{k^{\alpha}} \\
& = (\delta+\pGro)k^{1-\alpha}
\end{aligned}\end{gathered} s = 1 − k α k α − ( δ + γ ) k = ( δ + γ ) k 1 − α and the expression for the steady-state level of capital per capita
can be substituted in to yield
s = ( δ + γ ) ( ρ γ + ϑ + δ α ) − 1 = ( α ( γ + δ ) δ + ϑ + ρ γ ) \begin{gathered}\begin{aligned}
s & = (\delta+\pGro)\left(\frac{\CRRA \pGro+ \DiscRate + \delta}{\alpha}\right)^{-1}
\\ & = \left(\frac{\alpha(\pGro+\delta)}{\delta+\DiscRate+\CRRA \pGro}\right)
\end{aligned}\end{gathered} s = ( δ + γ ) ( α ρ γ + ϑ + δ ) − 1 = ( δ + ϑ + ρ γ α ( γ + δ ) ) The derivative of this expression with respect to γ \pGro γ
d s d γ = ( α ( δ + ϑ + ρ γ ) − α ( γ + δ ) ρ ( δ + ϑ + ρ γ ) 2 ) = ( α ( δ ( 1 − ρ ) + ϑ ) ( δ + ϑ + ρ γ ) 2 ) . \begin{gathered}\begin{aligned}
\frac{ds}{d\pGro} & = \left(\frac{\alpha(\delta + \DiscRate + \CRRA \pGro)-\alpha
(\pGro+\delta) \CRRA}{(\delta+\DiscRate+\CRRA \pGro)^{2}}\right)
\\ & = \left(\frac{\alpha(\delta(1-\CRRA) + \DiscRate)}{(\delta+\DiscRate+\CRRA
\pGro)^{2}}\right)
.
\end{aligned}\end{gathered} d γ d s = ( ( δ + ϑ + ρ γ ) 2 α ( δ + ϑ + ρ γ ) − α ( γ + δ ) ρ ) = ( ( δ + ϑ + ρ γ ) 2 α ( δ ( 1 − ρ ) + ϑ ) ) . This will be positive if its numerator is positive, i.e. if
ρ δ < ϑ + δ ρ < 1 + ϑ / δ . \begin{gathered}\begin{aligned}
\CRRA \delta & < \DiscRate + \delta
\\ \CRRA & < 1+\DiscRate/\delta .
\end{aligned}\end{gathered} ρ δ ρ < ϑ + δ < 1 + ϑ / δ . A typical assumption is ϑ = . 04 \DiscRate = .04 ϑ = .04 δ = . 08 \delta = .08 δ = .08 ρ \CRRA ρ ρ \CRRA ρ