\(C_i = [C_{i1}^{\frac{(\eta -1)}{\eta}} + C_{i2}^{\frac{(\eta -1)}{\eta}} + C_{i3}^{\frac{(\eta -1)}{\eta}}]^{\frac{\eta}{\eta -1}}\)
Setting the lagrangian:
\(\mathcal{L} = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{\eta}{\eta -1}} + \lambda[S - P_1C_1 - P_2C_2 - P_3C_3]\)
FOC:
\(\frac{\partial \mathcal{L}}{\partial C_1} = 0 = \frac{\eta}{\eta -1}[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*\frac{\eta -1}{\eta}C_1^{-\frac{1}{\eta}} + \lambda P_1\)
\(-\lambda P_1 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_1^{-\frac{1}{\eta}}\)
\(-\lambda P_2 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_2^{-\frac{1}{\eta}}\)
\(-\lambda P_3 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_3^{-\frac{1}{\eta}}\)
Therefore we have:
\(\frac{-\lambda P_1}{-\lambda P_2} = \frac{[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_1^{-\frac{1}{\eta}}}{[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_2^{-\frac{1}{\eta}}} = \frac{C_1}{C_2}^{-\frac{1}{\eta}}\)
\(C_2 = (\frac{P_1}{P_2})^{\eta}C_1\)
Doing the same thing for \(C_3\) yields:
\(C_3 = (\frac{P_1}{P_3})^{\eta}C_1\)
Since the constraint is equal to: \(S=P_1C_1 + P_2C_2 + P_3C_3\) we have:
\(S=P_1C_1 + P_2(\frac{P_1}{P_2})^{\eta}C_1 + P_3(\frac{P_1}{P_3})^{\eta}C_1\)
\(\frac{S}{P_1^\eta} = P_1^{1-\eta}C_1 + P_2^{1-\eta}C_1 + P_3^{1-\eta}C_1\)
\(C_1 = P_1^{-\eta}[\frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}]\)
Which we simply as follows \(C_1 = AP_1^{-\eta}\) where \(A = \frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}\)
Hence the elasticity of demand with respect to price for each good is equal to:
\(\frac{\partial C_1}{\partial P_1} = -\eta P_1^{-\eta - 1}[\frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}] - [\frac{(1-\eta)P_1^{-2\eta}}{(P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta})^2}]\)
\(\frac{P_1}{C_1} = P_1^{1+\eta}[\frac{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}{S}]\)
\(\frac{\partial C_1}{\partial P_1} \frac{P_1}{C_1} = -\eta - [\frac{(1-\eta)P_1^{1-\eta}}{S(P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta})}]\)
Where the first part of the equation is the substitution effect and the second part takes into account the income effect.
Since \(C = [C_1^{\frac{(\eta -1)}{\eta}} + C_2^{\frac{(\eta -1)}{\eta}} + C_3^{\frac{(\eta -1)}{\eta}}]^{\frac{(\eta -1)}{\eta}}\) and \(C_1 = AP_1^{-\eta}\) we have:
\(C = [(AP_1^{-\eta})^{\frac{(\eta -1)}{\eta}} + (AP_2^{-\eta})^{\frac{(\eta -1)}{\eta}} + (AP_3^{-\eta})^{\frac{(\eta -1)}{\eta}}]^{\frac{(\eta -1)}{\eta}}\)
\(C = A[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1-\eta}]^{\frac{(\eta -1)}{\eta}}\)
\(C = \frac{S[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1-\eta}]^{\frac{\eta -1}{\eta}}}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}\)
\(C = \frac{S}{[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}]^{\frac{1}{1-\eta}}}\)
This last equation just implies that the cost of obtaining one unit of C is equal to \(P=[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}]^{\frac{1}{1-\eta}}\). Where P is the price index. Hence we have:
\(A = \frac{S}{P^{1-\eta}}\)
\(C_i = AP_i^{-\eta}\)
\(C_i = S\frac{P_i^{-\eta}}{P^{1-\eta}}\)
\(C_i = \frac{S}{P}(\frac{P_i}{P})^{-\eta}\) and since \(C = \frac{S}{P}\)