Calculating the Competitive Equilibrium in a pure exchange economy with 3 comodities and 2 agents Consider a pure exchange economy with three commodities and two households with individual endowments
$e_{1}=(1,2,3) \text { and } e_{2}=(3,2,1)$
respectively, and utility functions
$u_{1}\left(x_{11}, x_{12}, x_{13}\right)=x_{11}+2 x_{12}+3 x_{13} \text { and } u_{2}\left(x_{21}, x_{22}, x_{23}\right)=3 x_{21}+2 x_{22}+x_{23} $
respectively.
Which of the following is the competitive equilibrium price vector?
Options:
(A) (3,2,3)
(B) (1,1,1)
(C) (1,2,1)
(D) None of the above
My attempt:
First I tried solving the optimization problem of the first agent
$ max \hspace{0.5 cm} u_{1}\left(x_{11}, x_{12}, x_{13}\right)=x_{11}+2 x_{12}+3 x_{13} \\ \text{subject to} \hspace{0.5 cm} p_{1}x_{1} + p_{2}x_{2} + p_{3}x_{3} = 3p_{1} + 2p_{2} + p_{3}\\
 $
I took three cases:
Case 1:
$
\frac{p_{1}}{1} &gt; \frac{p_{3}}{3} &gt; \frac{p_{2}}{2} \hspace{0.5 cm}  \text{or} \hspace{0.5 cm}  \frac{p_{1}}{1} &gt; \frac{p_{2}}{2} &gt; \frac{p_{3}}{3} \\  \text{Here I took the price level of a good i and divided it with the marginal utility of that } \\ \text{good. From this I concluded that the optimal bundle in this case should be:}   \hspace{0.5 cm} (0,\frac{p_{1}}{p_{2}} + 2 + 3\frac{p_{3}}{p_{2}},0) \hspace{0.5 cm} \\ \text{Now I took other cases as well where I took  the price relation as follows:} \hspace{0.5 cm}  $
$ \frac{p_{3}}{3} &gt; \frac{p_{2}}{2} &gt; \frac{p_{1}}{1} $
But the problem becomes very complicated and difficult to solve. Can somebody guide
First, there is no such thing as the competitive equilibrium price vector. If $(p_1,p_2,p_3)$ is a competitive equilibrium price vector, so is every positive multiple of this vector.
Second, to find the demand you also have to look at the cases where some inequalities are not strict, where they are actually equalities.
Here is how you can simplify the problem: The equilibrium consumption must be at least as good as keeping the endowment for both consumers. You can check that this means that if one of the consumers consumes the total endowment of a single good and nothing else, they are worse off. So in equilibrium, everyone must consume a positive quantity of at least two commodities. It is also easy to check that for no price system would both consumers consume positive amounts of all three commodities. There is no relative price $p_1/p_3$ under which both consumers consume positive amounts of both good $1$ and $3$. Togehter, this should help you pin down the (here unique) relative equilibrium prices.