Edgeworth Box (Non-Convex preference)

Consider a situation that agent A's indifference curves are concave, while B’s indifference curves are convex and both sets of indifference curves have exactly the same shape. A northeast movement increases A’s utility but decreases B’s utility.

I know U(x,y)= min(x,y) and U(x,y)= max{x,y} represent this case. But is there any other combination of utility functions that also represents that situation? Thank you so much.

If total endowments are $$(\bar x,\bar y)$$ and $$A$$ has utility function $$u_A(x_A,y_A)$$, then $$B$$'s utility function $$u_B(x_B,y_B)\equiv -u_A(\bar x-x_B,\bar y-y_B)$$ has the same indifference curves as $$A$$'s.
• This construction does not satisfy the condition that a movement in the northeast direction increases A's utility but decreases B's utility. The following correction will do the job: $u_B(x_B, y_B) = -u_A(\overline{x}-x_B, \overline{y}-y_B)$
• @KKecon, just use e.g. $u_A(x_A,y_A)=x^2_A+y^2_A$ with $(\bar x,\bar y)=(5,5)$ and substitute. Dec 13, 2022 at 16:38
• Thanks. So, for agent B' utility function, is it equal to $u_B(x_B, y_B) = - (\overline{x}-x_B)^2 - (\overline{y}-y_B)^2$ ? Dec 13, 2022 at 20:01