Y and Z are Pareto improvements on X, and they are also Pareto optimal. Points on the line between X and Y are also Pareto improvements on X but not Pareto opt

Y and Z are Pareto improvements on X, and they are also Pareto optimal. Points on the line between X and Y are also Pareto improvements on X but not Pareto optimal. R and U are not Pareto improvements on X, because as you note some people lose happiness, but they are Pareto optimal, since according to the model, no further Pareto improvements can be made on those positions. The point of the article, as I understand it, is that everyone would prefer Pareto improvements, but these are often infeasible. Economists seem to prefer Pareto optimums over non-optimums even when they are not Pareto improvements, perhaps because the word "optimum" and "efficient" have such positive suggestions. I think the curve is sort of relevant to making this point, but the harder I think about it, the more wrong it seems. It's meant to be (I guess) a theoretical model for what different public policies can achieve. But we would all prefer points between Y and Z to X, so why don't we enact policies that achieve that? If those policies are impossible or undiscovered, we should instead redraw the curve of optimality to reflect that, i.e. it should connect R, X, and U.