Gross Saving and Growth in the RCK Model

In the neoclassical growth model with labor-augmenting technological

progress at rate $\pGro$ , utility function $\util(c) = c^{1-\CRRA}/(1-\CRRA)$ ,

time preference rate $\DiscRate$ and depreciation rate $\delta$ the

steady-state will be at the point where the growth rate of consumption

is equal to the growth rate of labor-augmenting technological

progress, $\pGro$ ,

\begin{gathered}\begin{aligned} \frac{\dot{C}}{C} & = \CRRA^{-1}(f^{\prime}(k)-\delta-\DiscRate) \\ & = \pGro \end{aligned}\end{gathered}

which implies that

\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}

The aggregate gross saving rate is defined as

\begin{gathered}\begin{aligned} s & = \frac{y-c}{y} \\ & = \frac{k^{\alpha}-c}{k^{\alpha}} \\ & = 1-c/k^{\alpha}. \end{aligned}\end{gathered}

In steady-state by definition

\begin{gathered}\begin{aligned} \dot{k} & = 0 \end{aligned}\end{gathered}

but from the capital accumulation equation we know that

\begin{gathered}\begin{aligned} \dot{k} & = k^{\alpha}-(\delta+\pGro)k - c \end{aligned}\end{gathered}

so in steady-state

\begin{gathered}\begin{aligned} c & = k^{\alpha}-(\delta+\pGro)k. \end{aligned}\end{gathered}

This can be substituted into (3) to obtain

\begin{gathered}\begin{aligned} s & = 1-\frac{k^{\alpha}-(\delta+\pGro)k}{k^{\alpha}} \\ & = (\delta+\pGro)k^{1-\alpha} \end{aligned}\end{gathered}

and the expression for the steady-state level of capital per capita

can be substituted in to yield

\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}

The derivative of this expression with respect to $\pGro$ is

\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}

This will be positive if its numerator is positive, i.e. if

\begin{gathered}\begin{aligned} \CRRA \delta & < \DiscRate + \delta \\ \CRRA & < 1+\DiscRate/\delta . \end{aligned}\end{gathered}

A typical assumption is $\DiscRate = .04$ and $\delta = .08$ , implying that

the steady-state relationship between saving and growth in the neoclassical

model is positive only if the coefficient of relative risk aversion $\CRRA$

is less than 1.5. Typically we assume values of $\CRRA$ in the range from

2 to 5, so the model leads us to expect a negative relationship between saving

and growth.