# How was the CES production function derived?

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

# My question:

Though this is a quite elegant formula, how is it derived?

• I am perplexed by your question. What do you mean by "how is it derived"? Feb 1, 2018 at 20:57
• @denesp: Perhaps the OP wanted to ask who first proposed this functional form? Feb 1, 2018 at 21:39
• @HerrK. Let us hope for clarification, as that is a very different question. Feb 1, 2018 at 23:11
• His question makes sense - he is asking for the derivation of the functional form from its defining condition (i.e., constant elasticity of substitution).
– Ben
Feb 2, 2018 at 0:46

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.

• Would you mind showing how you derived the elasticity? Its a shame ODE's aren't a requirement for Statisitcs and Economics students because it seems quite essential as demonstrated in the answer to fully understand Feb 2, 2018 at 1:16