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\begin{document}

\title{Consumers Problem with Many Goods}
\maketitle
The consumer's maximization problem is:
\begin{equation}\label{EqUtilMax}
  \begin{split}
    &U_{j,s,t} = \sum_{u=0}^{E+S-s}\beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] u\left(c_{j,s+u,t+u},n_{j,s+u,t+u},b_{j,s+u+1,t+u+1}\right) \\
    &\text{where}\quad \rho_{s-1}=0 \\
    &\text{and} \quad u\left(c_{j,s,t},n_{j,s,t},b_{j,s+1,t+1}\right) = \frac{\left(c_{j,s,t}\right)^{1-\sigma} - 1}{1-\sigma} ... \\
    &\qquad\qquad + e^{g_y t(1-\sigma)}\chi^n_s\left(b\left[1 - \left(\frac{n_{j,s,t}}{\tilde{l}}\right)^\upsilon\right]^\frac{1}{\upsilon} + k\right) + \rho_s\chi^b\frac{\left(b_{j,s+1,t+1}\right)^{1-\sigma} - 1}{1-\sigma} \\
    &\text{and} \quad c_{j,s,t} = \prod_{i=1}^I \left( c_{i,j,s,t} - \bar c_{i,s} \right) ^{\alpha_i}; \quad \sum_{i=1}^I \alpha_i = 1 \\
    &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\forall j,t\quad\text{and}\:E+1\leq s\leq E+S
  \end{split}
\end{equation}

They maximize subject to the following budget constraint.
\begin{equation}\label{EqBC}
  \begin{split}
    \sum_{i=1}^I p_{i,t}c_{i,j,s,t} + b_{j,s+1,t+1} \leq \left(1 + r_t\right) b_{j,s,t} + w_t e_{j,s}&n_{j,s,t} + \frac{BQ_{j,t}}{\lambda_j\tilde{N}_t} - T_{j,s,t} \\
    \quad\text{where}\quad b_{j,s,1} = 0 \\
    &\text{for} \quad E+1\leq s \leq E+S \quad \forall j,t
  \end{split}
\end{equation}

We set up a Lagrangian and solve by taking derivatives with respect to $\{c_{i,j,s,t},n_{j,s,t+u},b_{j,s,t+1}\}$ for all $i,j,s$ and $t$.

With respect to each consumption good $i$:
\begin{equation}\label{Eqcfoc}
  \begin{split}
  \frac{\partial U}{\partial c_{i,j,s+u,t+u}} & = \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] \left[ \prod_{i=1}^I \left( c_{i,j,s,t} - \bar c_{i,s} \right) ^{\alpha_i} \right]^{-\sigma}\alpha_i \left( c_{i,j,s,t} - \bar c_{i,s} \right)^{\alpha_i-1} \\
   & - \lambda_{t+u} \left( p_{i,t+u} + \frac{\partial T_{j,s+u,t+u}}{\partial c_{i,j,s+u,t+u}} \right)= 0
    \end{split}
\end{equation}

With respect to labor:
\begin{equation}\label{Eqnfoc}
  \begin{split}
  \frac{\partial U}{\partial n_{j,s+u,t+u}} & = \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] e^{g_y (t+u)(1-\sigma)}\chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)^{v-1}\Biggl[1 - \biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)\Biggr]^{\frac{1-v}{v}} \\
  & - \lambda_{t+u} \left( w_{+u} e_{j,s+u} - \frac{\partial T_{j,s+u,t+u}}{\partial n_{j,s+u,t+u}} \right)= 0
    \end{split}
\end{equation}

With respect to savings:
\begin{equation}\label{Eqbfoc}
  \begin{split}
  \frac{\partial U}{\partial b_{j,s+u+1,t+u+1}} & = \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] \rho_s\chi^b\bigl(b_{j,s+U+1,t+U+1}\bigr)^{-\sigma} \\
  & - \lambda_{t+u} - \lambda_{t+u+1} \left( 1 + r_{t+u+1} - \frac{\partial T_{j,s+U+1,t+U+1}}{\partial b_{j,s+U+1,t+u+1}} \right)= 0
  \end{split}
\end{equation}

We can solve each of these for $\lambda_{t+u}$ to get the following.
\begin{equation}
  \begin{split}
  \lambda_{t+u} = \frac{ \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] \left[ \prod_{i=1}^I \left( c_{i,j,s,t} - \bar c_{i,s} \right) ^{\alpha_i} \right]^{-\sigma}\alpha_i \left( c_{i,j,s,t} - \bar c_{i,s} \right)^{\alpha_i-1} } { p_{i,t+u} + \frac{\partial T_{j,s+u,t+u}}{\partial c_{i,j,s+u,t+u}} } \nonumber
  \end{split}
\end{equation}

\begin{equation}
  \begin{split}
  \lambda_{t+u} = \frac{ \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] e^{g_y (t+u)(1-\sigma)}\chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)^{v-1}\Biggl[1 - \biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)\Biggr]^{\frac{1-v}{v}} } { w_{+u} e_{j,s+u} - \frac{\partial T_{j,s+u,t+u}}{\partial n_{j,s+u,t+u}} }  \nonumber
  \end{split}
\end{equation}

\begin{equation}
  \begin{split}
  \lambda_{t+u} = \beta^u\left[\prod_{v=s-1}^{s+u-1}(1-\rho_v)\right] \rho_s\chi^b\bigl(b_{j,s+U+1,t+U+1}\bigr)^{-\sigma} - \lambda_{t+u+1} \left( 1 + r_{t+u+1} - \frac{\partial T_{j,s+U+1,t+U+1}}{\partial b_{j,s+U+1,t+u+1}} \right)
    \end{split}  \nonumber
\end{equation}

These then reduce to the following $I+1$ Euler equations for each $j,s$ and $t$:

Marginal utility of consumption for each good $i$ compared to the marginal utility of labor:
\begin{equation}\label{EqcEuler}
  \begin{split}
  & \frac{ \left[ \prod_{i=1}^I \left( c_{i,j,s,t} - \bar c_{i,s} \right) ^{\alpha_i} \right]^{-\sigma}\alpha_i \left( c_{i,j,s,t} - \bar c_{i,s} \right)^{\alpha_i-1} } { p_{i,t+u} + \frac{\partial T_{j,s+u,t+u}}{\partial c_{i,j,s+u,t+u}} } \\
  & = \frac{ e^{g_y (t+u)(1-\sigma)}\chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)^{v-1}\Biggl[1 - \biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)\Biggr]^{\frac{1-v}{v}} } { w_{t+u} e_{j,s+u} - \frac{\partial T_{j,s+u,t+u}}{\partial n_{j,s+u,t+u}} }
   \end{split}
\end{equation}

Intertemporal Euler equation for savings, including the utility effects of bequests:
\begin{equation}\label{EqbEuler}
  \begin{split}
  & \frac{ e^{g_y (t+u)(1-\sigma)}\chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)^{v-1}\Biggl[1 - \biggl(\frac{n_{j,s+u,t+u}}{\tilde{l}}\biggr)\Biggr]^{\frac{1-v}{v}} } { w_{t+u} e_{j,s+u} - \frac{\partial T_{j,s+u,t+u}}{\partial n_{j,s+u,t+u}} } = \rho_s\chi^b\bigl(b_{j,s+U+1,t+U+1}\bigr)^{-\sigma}\\ 
  &  - \frac{ \beta(1-\rho_{s+u}) e^{g_y (t+u+1)(1-\sigma)}\chi^n_{s}\biggl(\frac{b}{\tilde{l}}\biggr)\biggl(\frac{n_{j,s+u+1,t+u+1}}{\tilde{l}}\biggr)^{v-1}\Biggl[1 - \biggl(\frac{n_{j,s+u+1,t+u+1}}{\tilde{l}}\biggr)\Biggr]^{\frac{1-v}{v}} } { w_{t+u+1} e_{j,s+u+1} - \frac{\partial T_{j,s+u+1,t+u+1}}{\partial n_{j,s+u+1,t+u+1}} } \times \\
  & \left( 1 + r_{t+u+1} - \frac{\partial T_{j,s+U+1,t+U+1}}{\partial b_{j,s+U+1,t+u+1}} \right)
  \end{split}
\end{equation}

% The aggrgeate consumption good is defined as follows.
% \begin{equation} \label{Eqcagg}
%     c_{j,s,t}  = \prod_{i=1}^I \left( c_{i,j,s,t} - \bar c_{i,s} \right) ^{\alpha_i} 
% \end{equation}

% The price of this aggregate good is given by the following.
% \begin{equation} \label{Eqpagg}
%     p_{j,s,t}  = \sum_{i=1}^I p_{i,t} c_{i,j,s,t} 
% \end{equation}

An Euler equation that compares marginal utilities of two arbitrary goods ($n$ \& $m$) is given below.
\begin{equation}\label{EqcEuler2}
  \frac{ \alpha_n \left( c_{n,j,s,t} - \bar c_{n,s} \right)^{\alpha_n-1} } { p_{n,t+u} + \frac{\partial T_{j,s,t}}{\partial c_{n,j,s,t}} } = \frac{ \alpha_m \left( c_{m,j,s,t} - \bar c_{m,s} \right)^{\alpha_m-1} } { p_{m,t+u} + \frac{\partial T_{j,s,t}}{\partial c_{m,j,s,t}} }
\end{equation}

We can use this equation for $m \in \{1,2,...,I\}$ solving for $c_{m,j,s,t} - \bar c_{m,s}$.
\begin{align}
    c_{m,j,s,t} - \bar c_{m,s} & = \left[ \left(c_{n,j,s,t} - \bar c_{n,s} \right)^{1-\alpha_n} \frac{\Gamma_m}{\Gamma_n} \right]^{\frac{1}{1-\alpha_m}}  \label{Eqcmdef} \\
    \Gamma_m & \equiv \frac{ \alpha_m } { p_{m,t} + \frac{\partial T_{j,s,t}}{\partial c_{m,j,s,t}} } \nonumber
\end{align}

% We then substitute this into equations \eqref{Eqcagg} and \eqref{Eqpagg} to get:

% \begin{equation} \label{Eqcagg2}
%   \begin{split}
%     c_{j,s,t} & \equiv \prod_{i=1}^I \left[ \left(c_{1,j,s,t} - \bar c_{1,s} \right)^{1-\alpha_1} \frac{\Gamma_i}{\Gamma_1} \right]^{\frac{\alpha_i}{1-\alpha_i}}  \\
%   \end{split}  
% \end{equation}

% \begin{equation} \label{Eqpagg2}
%     p_{j,s,t}  = \sum_{i=1}^I p_{i,t} \left( \left[ \left(c_{1,j,s,t} - \bar c_{1,s} \right)^{1-\alpha_1} \frac{\Gamma_i}{\Gamma_1} \right]^{\frac{1}{1-\alpha_i}} + \bar c_{i,s} \right) 
% \end{equation}

We can solve the household's problem fairly rapidly, if the values of $\frac{\partial T_{j,s,t}}{\partial c_{i,j,s,t}}$ are just constants, as they would be with a typical sales tax.
\begin{itemize}
\item First, given $\{p_{i,t}\}_{i=1}^I$ use Euler equation \eqref{EqcEuler} to find the value for $c_{1,j,s,t}$.
\item Then use equation \eqref{Eqcmdef} to get the rest of the $c_{i,j,s,t}$'s.
\end{itemize}

\end{document}