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The Constant Elasticity of Substitution production function is defined as:
(Taken from Wikipedia)
$$Q=F \boldsymbol{\cdot}\left(a\boldsymbol{\cdot}K^r+(1-a)\boldsymbol{\cdot}L^r \right)^{1\over{r}}$$
Where:
$Q=$ the quantity of output
$F=$ factor productivity
$a=$ Share parameter (i.e. $0 < a <1$)
$K, L=$ Quantities of production factors
$r= {\left(s-1 \right)\over{s}}$
$s= {1\over{1-r}}=$ Elasticity of subsitution
Though this is a quite elegant formula, how is it derived?
The CES function can be derived directly from the condition of constant elasticity of substitution. There are various ways to do this, but the simplest derivation occurs for a homothetic production function. Suppose we start with a homothetic production function $Q = f^*(K, L)$ and we rewrite this in intensive form as:
$$\begin{matrix} q = f(k) & & q \equiv Q/L & & k \equiv K/L. \end{matrix}$$
For this case the elasticity of substitution $s$ can be shown to be:
$$s = - \frac{f'(k)(f(k) - kf'(k))}{kf(k)f''(k)}.$$
Letting $r \equiv (s-1)/s$ and re-arranging this equation gives the second-order differential equation:
$$\frac{kf(k)f''(k)}{1-r} + f'(k)(f(k) - kf'(k)) = 0.$$
This equation has general solution $q = f(k) = c_0 (1 + c_1 k^r)^{1/r}$ where $c_0$ and $c_1$ are constants. Parameterising with $a \equiv c_1$ and $F \equiv c_0 \cdot c_1^{1/r}$ and substituting to obtain the extensive form gives:
$$\begin{equation} \begin{aligned} Q = Lq = Lf(K/L) &= c_0 L \left( \left( 1 + c_1 \frac{K}{L} \right)^r \right)^{1/r} \\ &= c_0 \left(c_1 K^r + L^r \right)^{1/r} \\ &= F \left(a K^r + (1-a) L^r \right)^{1/r}. \end{aligned} \end{equation}$$
The parameter $a$ can be interpreted as the capital intensity in production and the parameter $F$ can be interpreted as the overall efficiency of production.
