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.
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).
The function \(F\) is homogeneous of degree one.
For any \(\lambda>0\), we have:
\[ F(\lambda K,\lambda L) = \left(a \lambda^{\gamma}K^{\gamma}+(1-a)\lambda^{\gamma}L^{\gamma}\right)^{\frac{1}{\gamma}} = \lambda\left(a K^{\gamma}+(1-a)L^{\gamma}\right)^{\frac{1}{\gamma}} = \lambda F(K,L) \]
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: \[ f(k) = \left(a k^{\gamma}+1-a\right)^{\frac{1}{\gamma}} \]
The marginal products are positive.
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:
\[ F_L(K,L) = (1-a)\left(ak^{\gamma}+1-a\right)^{\frac{1-\gamma}{\gamma}}\geq 0 \]
The marginal products are decreasing.
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).
The elasticity of substitution between the factors is constant.
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.
The elasticity of output with respect to capital is not constant.
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:
\[ \epsilon_{Y/K}(k) = \frac{a \left(a +(1-a)k^{-\gamma}\right)^{\frac{1-\gamma}{\gamma}}}{\left(a + (1-a)k^{-\gamma}\right)^{\frac{1}{\gamma}}} \]
\[ \Leftrightarrow \epsilon_{Y/K}(k) = \frac{a}{a + (1-a)k^{-\gamma}} \]
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*}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 growth1.
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 \(k>0\).
Footnotes:
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 \(\sigma=1\))