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In an Edgeworth economy with two agents and two goods, Player A has endowment $\left(a,\,0\right)$ and utility $u_{A}\left(x_{1},\,x_{2}\right)=\min\left(\gamma x_{1},\,x_{2}\right)$ in which $\gamma>1$, and Player B has endowment $\left(0,\,b\right)$ and utility $u_{B}\left(x_{1},\,x_{2}\right)=\min\left(x_{1},\,\gamma x_{2}\right)$. When $a=b$, the competitive equilibrium is $$\left(x_{1}^{A},\,x_{2}^{A},\,x_{1}^{B},\,x_{2}^{B}\right)=\left(\frac{a}{1+\gamma},\,\frac{\gamma a}{1+\gamma},\,\frac{\gamma a}{1+\gamma},\,\frac{a}{1+\gamma}\right),\;\frac{p_{2}}{p_{1}}=1,$$ and A's utility at the equilibrium is $\frac{\gamma a}{1+\gamma}=\frac{\gamma b}{1+\gamma}$. However, when $a>b$, the competitive equilibrium becomes $$\left(x_{1}^{A},\,x_{2}^{A},\,x_{1}^{B},\,x_{2}^{B}\right)=\left(\frac{\gamma b-a}{\gamma^{2}-1},\,\frac{\gamma^{2}b-\gamma a}{\gamma^{2}-1},\,\frac{\gamma^{2}a-\gamma b}{\gamma^{2}-1},\,\frac{\gamma a-b}{\gamma^{2}-1}\right)$$ $$\frac{p_{2}}{p_{1}}=\frac{\gamma^{2}a-\gamma b}{\gamma^{2}b-\gamma a},$$ and A's utility at the equilibrium reduces to $\frac{\gamma^{2}b-\gamma a}{\gamma^{2}-1}<\frac{\gamma b}{1+\gamma}$. Is there any intuitive explanation for this reduction? (I'm not asking for solving the equilibrium because I already solved it.)

My proposed explanation is:

Because the supply of the first good increases from b to a, its price decreases, and the price of the second good increases. Therefore Player A must sell more $x_{1}$ to Player B, and Player B can sell less $x_{2}$ to Player A. Since both players have preferences with perfect complementation, the lesser amount of the two goods decides the utility, and Player A having less $x_{2}$ must have less utility.

Is this explanation appropriate?

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  • $\begingroup$ I guess it has something to do with the perfect complementation of the Leontief utility but simply don't know how to apply it. $\endgroup$ – Erik Apostol Apr 3 '17 at 2:59

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