Lump sum transfers to implement any Pareto efficient equilibrium as the market outcome

External question — Economics Stack Exchange Author: badeconomist101 Original post: https://economics.stackexchange.com/questions/58078 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. If we have 2 consumers (a and b) and 2 goods (x and y) -- so we are in an exchange economy setup. From what I understand, due to 2FWT, we can choose any Pareto efficient outcome (x*), calculate the corresponding relative prices, redistribute endowments to anywhere along the budget line, and then the consumers will trade to reach x*. However, how are we able to implement this if one of the goods cannot be traded? E.g. Imagine consumer a has an endowment 20 of x and 0 of y. Consumer b has an endowment of 0 of x and 12 of y. Consumer a has utility function: u = 2lnxa + 3lnya. Consumer b has utility function: u = 2lnxb + lnxb. We want to implement the Pareto efficient allocation (xa, ya) = (5, 6); (xb, yb) = (15, 6) using lump sum transfers but we are unable to trade good 2. How can we implement the Pareto efficient equilibrium using only transfers of good 1?

External answer — Economics Stack Exchange Author: Giskard Original post: https://economics.stackexchange.com/a/58079 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Here is the Edgeworth box of your exercise: The green dot with the curves running through it is your equilibrium allocation. The red and blue lines are indifference curves of $A$ and $B$ , while the orange line is the budget line. The other green dot in the lower right corner is the initial endowment. You want to move this to the budget line (orange line) by not trading the second good, so by moving only along the horizontal axis. Seems doable if you move this dot left by 7.5 units as your solution guide suggested.