# Why isn't this state a price equilibrium with transfer?

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.

• Who says that this isn't an equilibrium? Commented Dec 9, 2022 at 10:55

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.