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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.

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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.

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  • $\begingroup$ Thanks a lot. But could you provide an example to show this? $\endgroup$
    – KK econ
    Dec 13, 2022 at 12:31
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    $\begingroup$ 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: $u_B(x_B, y_B) = -u_A(\overline{x}-x_B, \overline{y}-y_B)$ $\endgroup$
    – Amit
    Dec 13, 2022 at 14:55
  • $\begingroup$ @Amit, that's right, sorry. Now corrected. $\endgroup$
    – VARulle
    Dec 13, 2022 at 16:35
  • $\begingroup$ @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. $\endgroup$
    – VARulle
    Dec 13, 2022 at 16:38
  • $\begingroup$ Thanks. So, for agent B' utility function, is it equal to $u_B(x_B, y_B) = - (\overline{x}-x_B)^2 - (\overline{y}-y_B)^2$ ? $\endgroup$
    – KK econ
    Dec 13, 2022 at 20:01

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