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This is probably a silly question, but I am misunderstanding something about the definition of a price equilibrium with transfers. Consider an Edgeworth Box (2 goods, 2 agents) consumer economy with both agents having utility function $U_a = U_b = x^2 + y^2$, and both having initial endowments $\omega_a = \omega_b = (2,2)$.

What is wrong with the following reasoning?

Consider the state $(x_a, y_a) = (4,3)$ and $(x_b, y_b) = (0,1)$. I argue that this is an equilibrium supported by prices $p = (1,1)$ and transfers $T_a = 3, T_b = -3$. First, agent $a$ wants to optimize $x^2 + y^2$ over the budget set $\{(x,y) \in [0,4]^2 : x + y \le 7\}$. This is achieved by the state $(4,3)$. Likewise $b$ wants to optimize $x^2 + y^2$ over their wealth of 1, and $(0,1)$ is one of the optimal states. Markets clear because this is an allocation.

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    $\begingroup$ Who says that this isn't an equilibrium? $\endgroup$
    – VARulle
    Dec 9, 2022 at 10:55

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Why must agent a select $x,y$ such that $x,y\leq 4$? An agent is allowed to select a bundle that is not feasible (outside of the box) if it is affordable at a given price, although the market will not clear at this price. His budget constraint is $\{(x,y): x,y\geq 0, x+y\leq 7\}$ and his demand is $(0,7)$ or $(7,0)$, clearly no equilibrium here.

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