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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} > \frac{p_{3}}{3} > \frac{p_{2}}{2} \hspace{0.5 cm} \text{or} \hspace{0.5 cm} \frac{p_{1}}{1} > \frac{p_{2}}{2} > \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} > \frac{p_{2}}{2} > \frac{p_{1}}{1} $

But the problem becomes very complicated and difficult to solve. Can somebody guide

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1 Answer 1

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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.

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  • $\begingroup$ I am not clear that why would the consumer be worse off if he consumes only one good and the entire endowment. Can you please help? Is it possible for you to send a brief working? I am not clear why would the agent consume at least two commodities in equilibrium. It would be of great help if you could give some more hint or a brief working. $\endgroup$
    – User_in
    Commented Apr 10, 2021 at 11:17
  • $\begingroup$ You calculate the utility of consumer $1$ when they consume their endowment $(1,2,3)$. Then you calculate the utility of consumer $1$ when they consume all of good $1$, which would be the consumption vector $(4,0,0)$. You do that for all commodities and for both consumers. Since the endowment is always in the budget set and a consumer in equilibrium always chooses a utility maximizing consumption bundle from the budget set, the chosen bundle must give as least as much utility as the endowment. $\endgroup$ Commented Apr 10, 2021 at 11:21

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