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Consider an economy with two agents. There are two goods, x and y. Agents' preferences are Leontif ones as follows: $u_1(x,y)=\min(x,4y)$ and $u_2(x,y)=\min(x,y)$. Initial endowment for 1 is (2,4), for 2 is (4,2) The problem asks me to use the excess demand function to find the equilibrium price and show that there are infinitely many equilibria with that price. The problem also asks what is the cause of such multiplicity of equilibria. I tried this many times, but I always get that the equilibirum price is (1,-1/2), which is impossible. Any help is appreciated.

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  • $\begingroup$ It looks like you are missing the endowment. Fix some equilibrium prices. What happens when you double both prices? $\endgroup$
    – Bayesian
    Nov 20 '20 at 12:47
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Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$.

In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$. And because of the preferences in optimum also $x_1 = 4 y_1$ and $x_2=y_2$ holds.

You use that to get 2's demand as $x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$ and a similar bur slightly different formula for $4 y^*_1 = x^*_1$.

The excess demand would be this demand minus the endowment. The sum of excess demands is zero in equilibrium.

In equilibrium, $x_1^*+x_2^* =2+4$ and $y_1^*+y_2^* =2+4$, which you can both use to calculate an equilibrium price ratio. Any price combination with that ratio clears the market.

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    $\begingroup$ and the rest is homework ;) $\endgroup$
    – Bayesian
    Nov 20 '20 at 15:11

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