SMAC (CES)

SMAC (CES)

The production function of the firm

$$q=A{\left[\alpha x_1^{-\rho }+(1-\alpha )x_2^{-\rho }\right]}^{\left(-\frac{1}{\rho }\right)}$$ $$\begin{array}{rl}f(\lambda x_1,\lambda x_2)& ={\left({\lambda }^{-\rho }\right)}^{\left(-\frac{1}{\rho }\right)}\cdot q\\ & =\lambda f(x_1,x_2)\end{array}$$ $$f_1=A\left[-\rho \alpha x_1^{-\rho -1}\right]\left(-\frac{1}{\rho }\right){\left[\alpha x_1^{-\rho }+(1-\alpha )x_2^{-\rho }\right]}^{\left(-\frac{1+\rho }{\rho }\right)}$$ $$f_2=A\left[-\rho (1-\alpha )x_2^{-\rho -1}\right]\left(-\frac{1}{\rho }\right){\left[\alpha x_1^{-\rho }+(1-\alpha )x_2^{-\rho }\right]}^{\left(-\frac{1+\rho }{\rho }\right)}$$ $$\text{MRTS}=\frac{f_1}{f_2}=\frac{\alpha x_1^{-\rho -1}}{(1-\alpha )x_2^{-\rho -1}}=\frac{\alpha }{1-\alpha }{\left(\frac{x_2}{x_1}\right)}^{(1+\rho )}$$ $$\begin{array}{rl}\rho & =\frac{d(x_2/x_1)}{x_2/x_1}÷\frac{d(\text{MRTS})}{\text{MRTS}}\\ & =\frac{d(x_2/x_1)}{x_2/x_1}÷\frac{\frac{\alpha }{1-\alpha }(1+\rho ){\left(\frac{x_2}{x_1}\right)}^{\rho }\cdot d(x_2/x_1)}{\frac{\alpha }{1-\alpha }{\left(\frac{x_2}{x_1}\right)}^{\rho +1}}\\ & =\frac{1}{1+\rho }\end{array}$$

So, $\boxed{\rho =\frac{1}{1+\rho }}$