<p>If total endowments are <span class="math-container">$(\bar x,\bar y)$</span> and <span class="math-container">$A$</span> has utility function <span class="math-container">$u_A(x_A,y_A)$</span>, then <span class="math-container">$B$</span>'s utility function <span class="math-container">$u_B(x_B,y_B)\equiv -u_A(\bar x-x_B,\bar y-y_B)$</span> has the same indifference curves as <span class="math-container">$A$</span>'s.</p>

CC BY-SA 4.0
2022-12-13T12:31:25.337Z
Thanks a lot. But could you provide an example to show this?
KK econ
CC BY-SA 4.0
2022-12-13T14:55:16.973Z
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: $u_B(x_B, y_B) = -u_A(\overline{x}-x_B, \overline{y}-y_B)$
Amit
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2022-12-13T16:35:19.673Z
@Amit, that's right, sorry. Now corrected.
VARulle
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2022-12-13T16:38:38.710Z
@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.
VARulle
CC BY-SA 4.0
2022-12-13T20:01:08.120Z
Thanks. So, for agent B' utility function, is it equal to $u_B(x_B, y_B) = - (\overline{x}-x_B)^2 - (\overline{y}-y_B)^2$ ?
KK econ
CC BY-SA 4.0
2022-12-14T08:20:13.240Z
@KKecon, yes. In this example you get $u_B(x_B,y_B)=10(x_B+y_B)-(x^2_B+y^2_B)-50$, but you can further simplify by discarding the constant.
VARulle
CC BY-SA 4.0
2022-12-13T10:31:04.213Z
2022-12-13T16:35:29.343Z
2022-12-13T16:35:29.343Z
VARulle
<p>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.</p>
<p>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.</p>

<microeconomics><general-equilibrium><pareto-efficiency><competitive-equilibrium><edgeworth-box>
Edgeworth Box (Non-Convex preference)
KK econ
# Edgeworth Box (Non-Convex preference)

(Asked by: username_0 on 2022-12-12)

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.

**Tags:** microeconomics, general-equilibrium, pareto-efficiency, competitive-equilibrium, edgeworth-box

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## 1 answers

### Answer by username_1 on 2022-12-13. Score: 2

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.

> **Comments:**
> **username_0**: Thanks a lot. But could you provide an example to show this?  
> **username_2**: 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: $u_B(x_B, y_B) = -u_A(\overline{x}-x_B, \overline{y}-y_B)$  
> **username_1**: @Amit, that's right, sorry. Now corrected.  
> **username_1**: @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.  
> **username_0**: Thanks. So, for agent B' utility function, is it equal to $u_B(x_B, y_B) = - (\overline{x}-x_B)^2 - (\overline{y}-y_B)^2$ ?  
> **username_1**: @KKecon, yes. In this example you get $u_B(x_B,y_B)=10(x_B+y_B)-(x^2_B+y^2_B)-50$, but you can further simplify by discarding the constant.  

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