Finding the set of Pareto Optimal Allocations - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2022-01-22T23:50:38Z https://economics.stackexchange.com/feeds/question/48152 https://creativecommons.org/licenses/by-sa/4.0/rdf https://economics.stackexchange.com/q/48152 0 Finding the set of Pareto Optimal Allocations user722271 https://economics.stackexchange.com/users/38457 2021-10-30T11:52:03Z 2021-10-31T16:10:17Z <p>I am asked to find the set of Pareto Optimal Allocations in an economy where there are two agents namely <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, with the following utility functions and endowments.</p> <p><span class="math-container">$$u_1({x_1}^1,{x_2}^2)= \beta log({x_1}^1)+(1-\beta)log({x_2}^2) \ ,\ {\omega}_1 = (0,1) \ \beta \in (0,1) \$$</span> <span class="math-container">$$u_2({x_2}^1,{x_2}^2)=min\{ {x_2}^1,{x_2}^2 \} \ ,\ {\omega}_2 = (1,0)$$</span></p> <p>I know that I can show the set of PO on the Edgeworth Box. However, how am I supposed to show the set of PO in algebraically? Shall I try splitting cases for the second agent's utility function?</p> <p>Thanks in advance.</p> <p><strong>Edit:</strong> Each agent <span class="math-container">$i$</span> has the preferences represented by the following utility function, <span class="math-container">$u_i$</span> and the endowment <span class="math-container">${\omega}_i$</span>. <span class="math-container">${x_i}^t$</span> denotes the amount of good <span class="math-container">$t$</span> consumed by the agent <span class="math-container">$i$</span>. Say, prices of the good 1 and good 2 are denoted by <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span>, respectively. I am asked to show the set of Pareto Optimal allocations in this setting.</p> https://economics.stackexchange.com/questions/48152/-/48175#48175 1 Answer by tdm for Finding the set of Pareto Optimal Allocations tdm https://economics.stackexchange.com/users/28130 2021-10-31T16:10:17Z 2021-10-31T16:10:17Z <p>Probably the easiest way to get all Pareto optimal allocations is to maxmize a weighted sum of utilities (of the two agents) subject to the resource constraints: <span class="math-container">$$\max_{x_1^1, x_1^2, x_2^1, x_2^2} \alpha u_1(x_1^1, x_1^2) + (1-\alpha) u_2(x_2^1, x_2^1) \text{ s.t. } x_1^1 + x_2^1 = 1 \text{ and } x_1^2 + x_2^2 = 1.$$</span> The PO allocations can be obtained by varying <span class="math-container">$\alpha$</span> over the interval <span class="math-container">$[0,1]$</span>.</p> <p>As the second agent has Leontief preferences any PO allocation will have <span class="math-container">$x_2^1 = x_2^2$</span>. So substuting out we have: <span class="math-container">$$\max_{x_2^1} \alpha u_1(1 - x_2^1, 1 - x_2^1) + (1-\alpha) u_2(x_2^1, x_2^1) \text{ s.t. } x_{2}^1 \le 1$$</span></p>