Consumer Demand in a Nested CES Utility Framework
Consider a consumer who has a two-layer nested utility function, where the upper nest is a Cobb-Douglas form:
\[U=\alpha \ln C_D+(1-\alpha) \ln C_M\]where
\[C_D=\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}}\]The consumption bundle \(C_D\) contains a number of varieties \(v\). Suppose the price level \(p\) is given. In the first step, we need to solve the upper nest:
\[\begin{array}{c} \max U=\alpha \ln C_D+(1-\alpha) \ln C_M \\ \text { s.t. } P_D C_D+P_M C_M=E \end{array}\]which gives the Marshallian demand for the composite goods \(C_D\) and \(C_M\) :
\[C_D=\frac{\alpha E}{P_D}, \quad C_M=\frac{(1-\alpha) E}{P_M}\]In the second step, we solve the demand for each variety \(v\) by maximizing the subutility \(C_D\) given the expenditure \(E_D=P_D C_D\) :
\[\begin{array}{l} \max C_D=\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}} \\ \text { s.t. } \sum p_v q_v=E_D \end{array}\]The first-order condition yields:
\[\begin{aligned} \lambda p_v & =\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]^{\frac{1}{\sigma-1}} b_v^{\frac{1}{\sigma}} q_v^{-\frac{1}{\sigma}} \\ & =\frac{C_D}{\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]} b_v^{\frac{1}{\sigma}} q_v^{-\frac{1}{\sigma}} \end{aligned}\]Then,
\[q_v=\frac{C_D^\sigma b_v}{\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]^\sigma p_v^\sigma} \lambda^{-\sigma}\]Multiply both sides by \(p_v\) and sum over \(v\), we can solve for \(\lambda\) :
\[\lambda^\sigma=\frac{C_D^\sigma\left[\sum_v p_v^{1-\sigma}\right]}{E_D\left[\sum_v b_v^{\frac{1}{\sigma}} q_v^{\frac{\sigma-1}{\sigma}}\right]^\sigma}\]Plug \(\lambda\) back into \(q_v\) :
\[q_v=\frac{b_v E_D}{p_v^\sigma P_D^{1-\sigma}}\]where \(E_D=\frac{\alpha E}{P_D}\) and \(P_D=\left[\sum_v b_v p_v^{1-\sigma}\right]^{\frac{1}{1-\sigma}}\).