Perfect substitutes preference relation with discontinuity

External question — Economics Stack Exchange Author: user584534 Original post: https://economics.stackexchange.com/questions/60331 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Consider a 2x2 exchange economy where both agents have a perfect substitute kind of preference $x_1+x_2$ but agent A prefers bundle with more of good 1 when the sum is the same and agent B prefers bundle with more of good 2 when the sum is equal. Firstly I would like to confirm that this preference relation is not continuous. Secondly, under what conditions on the endowments does a WE exist. And when this is satisfied, what are all the WE? According to my understanding, endowments need to be such that A gets $(w_1,0)$ and B gets $(w_2,0)$ . And no trade is the only WE and it holds for all positive prices. Please let me know if this is correct. If not, how would you approach this problem. Thank you!

External answer — Economics Stack Exchange Author: Lulu Original post: https://economics.stackexchange.com/a/60337 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. To show that the preference relation is not continuous, note that although preferences are defined by the sum $x_1 + x_2$ , they include a lexicographic tie-breaking rule. Within an indifference curve (i.e., for bundles where $x_1 + x_2$ is constant), agent A strictly prefers the bundle with a higher quantity of good 1. By the definition of continuity of preferences, if a sequence of bundles ${x^n}$ converges to $x$ , and $x^n \succeq y$ for all $n$ , then it must be that $x \succeq y$ . We construct a counterexample to this condition: Let $x^n = (x_1 + \frac{1}{n}, x_2 + 1)$ and $y = (x_1 + 1, x_2)$ . Then for all $n$ : $$ x^n_1 + x^n_2 = x_1 + \frac{1}{n} + x_2 + 1 > x_1 + 1 + x_2 = y_1 + y_2 $$ Thus, $x^n \succ_A y$ for all $n$ . But as $n \to \infty$ , $x^n \to x = (x_1, x_2 + 1)$ . Then: $$ x_1 + x_2 + 1 = y_1 + y_2 $$ So both bundles lie on the same indifference curve. Under agent A's preference, since the sums are equal, the bundle with more of good 1 is strictly preferred. But $x_1 < x_1 + 1$ , hence: $$ x \prec_A y $$ This contradicts the requirement that $x \succeq_A y$ , and thus, preferences are not continuous. For the second part, the Walrasian demand for each agent is obtained by solving: For agent A: $$\max_{x_1^A,x_2^A} x_1 + x_2$$ $\text{s.t. } x_1 + px_2 = w_1^A + pw_2^A$ FOC $1 - \lambda \leq 0$ $1 - p\lambda \leq 0$ (Kuhn-Tucker conditions) The Kuhn-Tucker conditions imply that A will choose the cheaper good, breaking ties in favor of good 1. Thus, A’s net demand depends on the price $p$ : $$ND^A = \begin{cases} \left(pw_2^A, -w_2^A\right) & \text{if } p> 1\\ \left(w_2^A,-w_2^A\right) & \text{if } p= 1\\ \left(-w_1^A, \frac{w_1^A}{p}\right) & \text{if } p<1\end{cases}$$ Analogously, agent B strictly prefers more of good 2 when $x_1 + x_2$ is equal, so B will always break ties in favor of good 2: $$ND^B = \begin{cases} \left(pw_2^B, -w_2^B\right)& \text{if } p > 1\\\left(-w_1^B,w_1^B\right) & \text{if } p = 1\\ \left(-w_1^B ,\frac{w_1^B }{p}\right)& \text{if } p < 1\end{cases}$$ To determine the existence of a Walrasian Equilibrium (WE), market clearing must hold: $$ND_1^A + ND_1^B =0 \implies ND_2^A + ND_2^B = 0$$ (Walras property) It is straightforward to verify that market clearing only occurs at $p = 1$ , and only under the condition: $$ w_2^A - w_1^B= 0 \implies w_2^A = w_1^B$$ This condition implies that the quantities each agent is willing to trade must be equal: agent A must want to sell exactly the amount of good 2 that agent B wants to buy, and vice versa for good 1. Since agent A is only willing to sell her endowment of good 2, and agent B can only purchase good 1 using his endowment of good 1, the equilibrium requires that the quantity A wants to sell equals the quantity B can offer in exchange. Therefore, a necessary condition for equilibrium is that $w_2^A = w_1^B$ . The endowments $w_1^A$ and $w_2^B$ do not affect the feasibility of the exchange because, at equilibrium, these quantities are not traded: each agent keeps their endowment of the good they do not wish to exchange.