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#+TITLE: Properties of the CES production function
#+DATE: October 2022
#+AUTHOR: Stéphane Adjemian
#+EMAIL: stephane.adjemian@univ-lemans.fr

#+BEGIN_QUOTE
In the lectures we study the Solow model with a Cobb-Douglas production
function. To appreciate why the various properties of neoclassical production
functions matter, it is useful to see how the predictions of the model change
with another production function. This note is a brief presentation of the
CES production function.
#+END_QUOTE

\\
\\
\\

The CES (for Constant Elasticity of Substitution) production function is
defined by:

\[
F(K, L) = \left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}}
\]

with $0 < a < 1$, $\gamma < 1$ and $\gamma\neq 0$. One can show, see below, that the
parameter $\gamma$ determines the elasticity of substitution between the
factors: \[ \sigma = \frac{1}{1-\gamma} \] which is constant (hence the name
of the production function). Depending on the value of the parameter
$\gamma$, this elasticity can therefore take any value between $0$ (the
factors are perfect complements) and $\infty$ (the factors are perfect
substitutes).\\


#+BEGIN_property
The function $F$ is homogeneous of degree one.
#+END_property

#+BEGIN_proof
For any $\lambda>0$, we have:

\[
F(\lambda K,\lambda L) = \left(a \lambda^{\gamma}K^{\gamma}+(1-a)\lambda^{\gamma}L^{\gamma}\right)^{\frac{1}{\gamma}} = \lambda\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}} = \lambda F(K,L)
\]
#+END_proof

The CES production function defined above therefore exhibits constant returns
to scale. It is thus possible to define the intensive production function,
expressing output per worker as a function of the capital stock per worker:
\[ f(k) = \left(a k^{\gamma}+1-a\right)^{\frac{1}{\gamma}} \]

#+BEGIN_property
The marginal products are positive.
#+END_property

#+BEGIN_proof
For the marginal product of capital, we have:

\begin{equation*}
\begin{split}
F_K(K,L) &= \frac{1}{\gamma} a \gamma K^{\gamma-1}\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}-1}\\
&= a K^{\gamma-1}\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}-1}\\
&= a K^{-\frac{1-\gamma}{\gamma}\gamma}\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1-\gamma}{\gamma}}\\
&= a \left(a +(1-a)k^{-\gamma}\right)^{\frac{1-\gamma}{\gamma}}\geq 0
\end{split}
\end{equation*}

The marginal product of capital depends only on the capital stock per worker:
it is homogeneous of degree zero, because the production function is
homogeneous of degree one. In the same way, for the marginal product of
labour, one shows that:

\[
F_L(K,L) = (1-a)\left(ak^{\gamma}+1-a\right)^{\frac{1-\gamma}{\gamma}}\geq 0
\]
#+END_proof

#+BEGIN_property
The marginal products are decreasing.
#+END_property

#+BEGIN_proof
The marginal product of capital is a decreasing function of the capital stock
per worker, $k$:

\[
\frac{\mathrm d F_K}{\mathrm d k} = -a(1-a)(1-\gamma)k^{-\gamma-1}\left(a+(1-a)k^{-\gamma}\right)^{\frac{1-\gamma}{\gamma}-1} < 0
\]

for any admissible value $\gamma < 1$. In the same way one shows that the
marginal product of labour is always an increasing function of the capital
stock per worker. Moreover, the capital stock per worker is an increasing
function of $K$ and a decreasing function of $L$. All in all, the marginal
product of capital (respectively of labour) is a decreasing function of
capital (respectively of labour).
#+END_proof

#+BEGIN_property
The elasticity of substitution between the factors is constant.
#+END_property

#+BEGIN_proof
The isoquant of level $\bar Y$ can be defined as the set of pairs $(L,K)$ such
that $F(K,L)=\bar Y$. Along an isoquant, we must therefore have:

\[
F_K(K,L)\mathrm dK + F_L(K,L)\mathrm d L = 0
\]

\[
\Leftrightarrow \frac{\mathrm dK}{\mathrm d L} =- \frac{F_L(K,L)}{F_K(K,L)}
\]

the slope of the isoquant of level $\bar Y$ at a point $(K,L)$, a.k.a. the
MRTS (up to the sign). For the CES function, we have:

\[
\textrm{MRTS}(K,L) = \frac{(1-a)\left(ak^{\gamma}+1-a\right)^{\frac{1-\gamma}{\gamma}}}{a \left(a +(1-a)k^{-\gamma}\right)^{\frac{1-\gamma}{\gamma}}}
\]

\[
\Leftrightarrow \textrm{MRTS}(K,L) = \frac{(1-a)}{a} k^{1-\gamma}
\]

\[
\Rightarrow \log \textrm{MRTS}(K,L) = \log\frac{(1-a)}{a} +(1-\gamma) \log k
\]

\[
\Rightarrow  \frac{\mathrm d \textrm{MRTS}}{\textrm{MRTS}} = (1-\gamma)\frac{\mathrm d k}{k}
\]

\[
\Leftrightarrow \underbrace{\frac{\frac{\mathrm d k}{k}}{\frac{\mathrm d \textrm{MRTS}}{\textrm{MRTS}}}}_{\sigma} = \frac{1}{1-\gamma}
\]

The elasticity of substitution between the factors $K$ and $L$, which we
denote $\sigma$, characterises the curvature of the isoquant by relating the
rate of change of the factor ratio, $k=\frac{K}{L}$, to the rate of change of
the slope of the isoquant.
#+END_proof

#+BEGIN_property
The elasticity of output with respect to capital is not constant.
#+END_property


#+BEGIN_proof
The elasticity of output with respect to the capital stock is defined by:

\[
\epsilon_{Y/K} = \frac{F_K(K,L)}{\frac{F(K,L)}{K}}
\]

the ratio of the marginal product to the average product. We expressed the
marginal product as a function of $k$ above; we can do the same for the
average product:

\begin{equation*}
\begin{split}
\frac{F(K,L)}{K} &= K^{-\frac{\gamma}{\gamma}}\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}}\\
&= \left(a + (1-a)\left(\frac{L}{K}\right)^{\gamma}\right)^{\frac{1}{\gamma}}\\
&= \left(a + (1-a)k^{-\gamma}\right)^{\frac{1}{\gamma}}\\
\end{split}
\end{equation*}

Hence:

\[
\epsilon_{Y/K}(k) = \frac{a \left(a +(1-a)k^{-\gamma}\right)^{\frac{1-\gamma}{\gamma}}}{\left(a + (1-a)k^{-\gamma}\right)^{\frac{1}{\gamma}}}
\]

\[
\Leftrightarrow \epsilon_{Y/K}(k) = \frac{a}{a + (1-a)k^{-\gamma}}
\]
#+END_proof

#+BEGIN_property
The elasticity of output with respect to capital is a monotonically increasing function of $k$ if $0 < \gamma < 1$, and monotonically decreasing if $\gamma < 0$. Moreover:
\begin{equation*}
\lim_{k\rightarrow\infty}\epsilon_{Y/K}(k) =
\begin{cases}
1, &\text{ if } 0 < \gamma < 1\\
0 &\text{ if } \gamma < 0
\end{cases}
\end{equation*}
#+END_property

When $\gamma>0$, that is $\sigma>1$, the production function becomes
asymptotically linear. This explains why, once embedded in the Solow model,
this production function may generate endogenous growth[fn:1: the model may
generate endogenous growth when the factors are more substitutable than in
the Cobb-Douglas case (we showed, in the lectures, that the elasticity of
substitution is then $\sigma=1$)].

\\

The CES production function can be interpreted as a generalisation of the
Cobb-Douglas production function. It suffices to note that when $\gamma$
approaches zero, $\sigma$ tends to one, and the $\mathrm{MRTS}$ of the CES
converges to the $\mathrm{MRTS}$ of the Cobb-Douglas for any $k>0$.