The neoclassical growth model is derived from the Cobb Douglas production function
$$Y_{t} = A K_{t}^\alpha L_{t}^{1-\alpha} $$
where
$$Y = Output$$
$$A = Total Factor Productivity$$
$$K = Capital$$
$$L = Labor$$
$$\alpha = Production \: elasticity \: of \: capital \: and \: capital \: income \: share \: under \: perfect \: competition$$
$$1-\alpha = Production \: elasticity \: of \: labor \: and \: labor \: income \: share \: under \: perfect \: competition$$
$$t=time$$
Because of linear homogeneity, which is that
$$mY_{t}= A (\, mK )\,_{t}^\alpha (\, mL )\,_{t}^{1-\alpha}$$
and setting m=1/L, it follows that
$$y_{t}=Ak_{t}^\alpha$$
where
$$y_{t}=output \: per \: worker$$
$$k_{t} = capital \:per\: worker$$
The production function is called “neo-classical” because it embeds the assumption of positive, but diminishing returns to capital per labor. This is,
$$ \frac{\partial y_{t}}{\partial k_{t}} = \alpha A k_{t}^{\alpha-1} > 0$$
$$ \frac{\partial y_{t}}{\partial k_{t}} = \alpha (\, \alpha - 1 )\, A k_{t}^{\alpha-2} > 0$$
The neo-classical production function needs to be distinguished from the Keynesian production function, which is linear.
The standard Keynesian growth model is the so-called Harrod-Domar model and assumes a linear relationship between capital and output (Van den Berg, 2001).
The problem with linear production functions, however, is that they only apply to short-run dynamics of unemployment.
The shape of a neoclassical production function is illustrated in Figure 1.
Figure 1: The Neoclassical Production Function
In order to see what dynamics the neoclassical production function follows over time, the derivative of the per capita production function with respect to time is taken. This is, $$\dot{y}_{t} = A f^\prime (k_{t}) \frac{d k_{t}}{d K_{t}} \frac{d K_{t}}{dt} + A f^\prime (k_{t}) \frac{d k_{t}}{d L_{t}} \frac{d L_{t}}{d t} = A f^\prime (k_{t}) (\frac{1}{L_{t}}\dot{K}_{t} - \frac{K_{t}}{L_{t}^2}\dot{L}_{t})$$ Since $$\dot{K}_{t}= s Y_{t} - \delta K_{t}$$ which is savings minus capital depreciation, where $$s= savings \: rate$$ $$\delta = capital\:depreciation\: rate$$ and $$\dot{L}_{t}=n L_{t}$$ which is the population growth rate, where $$n=population\:growth\:rate$$ it follows that $$\dot{y}_{t}=A f^\prime (k_{t})[sy_{t}-(n+\delta)k_{t}]$$ where $$A f^\prime (k_{t}) = marginal\: product \: of\:labor$$ Important is now the term \( [sy_{t}-(n+\delta)k_{t}] \), which is the so-called Solow equation, and from which the growth dynamics can be derived. They are illustrated in Figure 2.
Figure 2: Solow Growth Model
Output per worker grows whenever \( [sy_{t} > (n+\delta)k_{t}] \) and declines whenever \( [sy_{t} > (n+\delta)k_{t}] \). The Solow growth model suggests that GDP per capita inevitably reaches a steady state at which output per capita stays constant. This is the case at k*. Of course, more advanced economic growth models can circumvent the problem of a steady state. These are so-called growth models with exogenous technological progress or, more sophisticatedly, endogenous growth models.