Elasticity of substitution

The elasticity of substitution between two inputs of a production function (or two goods in a utility function) measures the percentage change in the ratio of the two inputs relative to the percentage change in their prices.

The elasticity of substitution represents the curvature of the isoquant, this is, the degree of substitutability between inputs. Mathematically, for inputs $x_{1}$ and $x_{2}$ and a production function $f(x_{1}, x_{2}),$ it is defined as:

$$\sigma_{1,2} = - \frac{\partial \frac{x_{1}}{x_{2}}}{\partial \frac{p_{1}}{p_{2}}} \frac{\frac{p_{1}}{p_{2}}}{\frac{x_{1}}{x_{2}}}.$$

Assuming perfect competition, factor prices equal their margial product: $p_{1} = \frac{\partial f(x_{1},x_{2})}{\partial x_{1}}.$ Hence:

$$\sigma_{1,2} = \textcolor{orange}{\boldsymbol{-}} \frac{ \partial \frac{x_{1}}{x_{2}}} {\partial \frac{ \frac{\partial f(x_{1},x_{2})}{\partial x_{1}}}{\frac{ \partial f(x_{1},x_{2})}{\partial x_{2}}}}
\frac{ \frac{ \frac{ \partial f(x_{1},x_{2})}{\partial x_{1}}}{ \frac{ \partial f(x_{1},x_{2})}{\partial x_{2}}}} { \frac{x_{1}}{x_{2}}} = \textcolor{orange}{\boldsymbol{-}} \frac{ \partial \ln \left( \frac{x_1}{x_2} \right)} {\partial \ln \left( \frac{ \frac{\partial f(x_1,x_2)}{\color{red}{\partial x_1}}}{\frac{ \partial f(x_1, x_2)}{\color{green}{\partial x_2}}} \right)} = \frac{ \partial \ln \left( \frac{x_1}{x_2} \right)} {\partial \ln \left( \frac{ \frac{\partial f(x_1,x_2)}{ \color{green}{\partial x_2}}}{\frac{ \partial f(x_1, x_2)}{\color{red}{\partial x_1}}} \right)} $$

Cobb-Douglas example

The Cobb-Douglas production function has an elasticity of substitution equal to one.

$$\begin{eqnarray} & F(K,L)= A K^\alpha L^{1-\alpha}, \\\ & w = A(1-\alpha) K^\alpha L^{-\alpha}, \\\ & r = A \alpha K^{\alpha-1} L^{1-\alpha}.\end{eqnarray}$$ Hence: $$\frac{w}{r} = \frac{1-\alpha}{\alpha} \frac{K}{L}.$$ Therefore, we can compute $\sigma_{K,L}$ as

$$\frac{ \partial \ln \left(\frac{K}{L}\right)}{\partial \ln \left(\frac{w}{r}\right)} = \frac{\color{red}{\partial \left( \frac{K}{L} \right)}}{\color{green}{\frac{K}{L}}} \frac{\frac{w}{r}}{\color{red}{\partial \frac{w}{r}}} = \color{red}{\underbrace{\frac{\alpha}{1-\alpha}}_{\frac{\partial \left( \frac{K}{L} \right)}{\partial \frac{w}{r}}}} \color{green}{\underbrace{\frac{1-\alpha}{\alpha}\frac{K}{L}}_{\left(\frac{w}{r}\right)}}\frac{L}{K}=1$$

CES function

In the case of a CES function, the elasticity of substitution equals $\frac{1}{1-\rho}.$

$$\begin{eqnarray} & F(K,L)= A \left( \alpha K^{\rho} + (1-\alpha) L^{\rho} \right)^\frac{\nu}{\rho}, \rho \leq 1 \\\ & w = A \frac{\nu}{\rho}\left( \alpha K^{\rho} + (1-\alpha) L^{\rho}\right)^{\frac{\nu}{\rho}-1} ( 1- \alpha) \rho L^{\rho-1}, \\\ & r = A \frac{\nu}{\rho}\left( \alpha K^{\rho} + (1-\alpha) L^{\rho} \right)^{\frac{\nu}{\rho}-1}\alpha \rho K^{\rho-1}.\end{eqnarray}$$ Hence: $$\frac{w}{r} = \frac{1-\alpha}{\alpha} \left(\frac{K}{L} \right)^{1-\rho}.$$

The parameter $\nu$ indicates the degree of homogeneity of the function.

The elasticity of substitution equals:

$$\frac{ \partial \ln \left(\frac{K}{L}\right)}{\partial \ln \left( \frac{w}{r} \right)} = \frac{ \partial \frac{K}{L}}{\frac{K}{L}}\frac{\frac{w}{r}}{\partial \frac{w}{r}} = \frac{\alpha}{1-\alpha}\frac{1}{1-\rho}\left(\frac{w}{r}\frac{\alpha}{1-\alpha}\right)^{\frac{1}{1-\rho}-1} \frac{1-\alpha}{\alpha}\left(\frac{K}{L}\right)^{1-\rho}\frac{L}{K} = \frac{1}{1-\rho}.$$

Cobb-Douglas and constant shares

The Cobb-Douglas production is the unique neoclassical production function that exhibits constant factor shares in a competitive setup. From the definition of shares we have that:

$$\frac{rK}{Y} = z_{1}$$ $$\frac{wL}{Y} = z_{2}$$ Where $z_1$ and $z_2$ are constants.

Starting from the first, and because of competititon: $\frac{rK}{Y} = z_{1} \implies \frac{rk}{y} = z_1 \implies \frac{f^\prime (k) k}{f(k)} = z_1.$ Solving this differential equation: $\frac{f^\prime (k) k}{f(k)} = z_{1} \implies \frac{\mathrm{d} \log (f(k))}{\mathrm{k}} = \frac{z_1}{k} \implies$ $\implies \log (f(k)) = z_{1} \log (k)+C\implies f(k) = Ck^{z_1}.$ Which corresponds to the intensive form of the Cobb-Douglas production function.