How Do We Get Cobb-Douglas and Leontief Functions from CES Function : A Lecture Note on Discrete and Continuum Differentiated Object Models

Most lectures teach the relationship between the CES, Cobb-Douglas, and Leontief functions using the value of elasticity of substitution, namely, in the discrete object model. This lecture note aims at being a reference for algebraic computations of the Leontief and Cobb-Douglas functions by taking limits of CES functions both in discrete and continuum goods models. The argument on the discrete case uses l'Hôpital's rule as usually done. The argument on the continuum case also uses l'Hôpital's rule to show the convergence to the Cobb-Douglas function. To guarantee the convergence to the Leontief function, however, we rely on the squeeze principle