Consider the following CES utility function for the $h$ household \begin{equation} U^h(x^h_1,\ldots,x^h_N) = \left[ \sum_{j=1}^{N} (x^h_j - \zeta^h_j)^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}} \end{equation} where $\sigma \in (0,\infty)$ denotes the elasticity of substitution and $\zeta^h_j \in [-\bar{\zeta}, \bar{\zeta}]$ denotes whether the particular good is a necessity or not for the household. For example, $\zeta^h_j >0$ may mean that household h needs to consume at least a certain amount of good $j$ to survive.

It is assumed that

\begin{equation} \sum_{j=1}^{N} p_j \bar{\zeta} < w^h \end{equation} where $w^h$ denotes the household $w^h$ income.

Now, to derive the indirect utility function, that according to the book reads \begin{equation} \nu^h(p,w^h)= \frac{\left[- \sum_{j=1}^{N} p_j \zeta^h_j + w^h \right]}{\left[\sum_{j=1}^{N} p^{1-\sigma}_j \right]^{\frac{1}{1-\sigma}}} \end{equation}

I should find the optimal consumption bundles and then plug them into the utility function. I proceeded as follows

Considering two commodities $x^h_j$ and $x^h_{j'}$ with $j \ne j'$: \begin{equation} \frac{\partial \mathcal{L}}{\partial x_j} \Bigg/ \frac{\partial \mathcal{L}}{\partial x_j'} =\left( \frac{{x^h_{j'}-\zeta^h_{j'}}}{{x^h_{j}-\zeta^h_{j}}}\right)^{\frac{1}{\sigma}} = \frac{p_j}{p_{j'}} \end{equation} where $\mathcal{L}$ denotes the lagrangian. The constraint I used is: $\sum_{j=1}^{N} p_j x_j = w^h$. Now, solving the expression above by $x^h_{j'}$ and plugging it into the constraint \begin{equation} \sum_{j \ne j'} p_j x^h_j + (p_j)^{\sigma} p^{1-\sigma}_{j'} (x^h_j - \zeta^h_j) + p_{j'} \zeta^h_{j'}=w^h \end{equation} I'm not sure that what I'm doing is correct. If so, how should I continue?


1 Answer 1


Fix some good $r$. For notational convenience, let me drop the supscript $h$. From the first order conditions, we get: $$ x_{j} = \zeta_j + \left(\frac{p_{r}}{p_{j}}\right)^{\sigma}(x_{r} - \zeta_{r}) $$ Substitute into the budget constraint $\sum_{j} p_j x_j = w$. $$ \begin{align*} &\sum_j p_j \left(\frac{p_r}{p_j}\right)^\sigma (x_r - \zeta_r) + \sum_j p_j \zeta_j = w,\\ \iff & \sum_j p_j^{1 - \sigma} p_r^\sigma (x_r - \zeta_r) = w - \sum_j p_j \zeta_j,\\ \iff & p_r^\sigma (x_r - \zeta_r) = \frac{w - \sum_j p_j \zeta_j}{\sum_j p_j^{1 - \sigma}},\\ \iff & x_r = \zeta_r + \frac{w- \sum_j p_j \zeta_j}{p_r^\sigma \sum_j p_j^{1 - \sigma}} \end{align*} $$

Now substitute this into the utility function $$\left(\sum_r (x_r - \zeta_r)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}. $$

We obtain: $$ \begin{align*} &\left(\sum_r \left(\frac{w - \sum_j p_j \zeta_j}{p_r^\sigma \sum_j p_j^{1 - \sigma}}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}},\\ =& \left[w - \sum_j p_j \zeta_j\right] \frac{\left(\sum_r p_r^{1 - \sigma}\right)^{\frac{\sigma}{\sigma - 1}}}{\sum_j p_j^{1 - \sigma}},\\ = & \frac{\left[w - \sum_j p_j \zeta_j\right]}{\left(\sum_r p_r^{1 - \sigma}\right)^{\frac{1}{1-\sigma}}},\\ \end{align*} $$


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