Consider points A, B, C, and D shown in the accompanying Edgeworth box diagram.

(i) Which of these points do Bart and Lisa both prefer to the endowment point O?

(ii) Which of these points are Pareto optimal?

(iii) Which of these points are competitive equilibria? Justify your answers.

(i) Points A and B are Pareto-preferred to point O, because both points would put Lisa and Bart on higher indifference curves, so both points lie in the region of mutual advantage. A movement to point C or D would make Bart worse off, so these points are not Pareto-preferred to the initial endowment.
(ii) Points B and C are Pareto optimal because they lie where two indifference curves are tangent and thus lie on the contract curve. At points A and D, indifference curves are intersecting, so other points are Pareto-preferred to these allocations.
(iii) Point B is a competitive equilibrium-when the relative price of food equals the slope of the line drawn through point O, both Lisa and Bart choose point B, so the market clears and a competitive equilibrium is achieved.