Question

Lisa’s family consumes the following commodities on three different days in a week:

• 8 garlic breads and 2 pizzas on Monday
• 5 garlic breads and 3 pizzas on Thursday
• 3 garlic breads and 4 pizzas on Friday

The family derives the same level of satisfaction from these combinations. Does this phenomenon ensure the convexity of Indifference curve? Justify your answer. 

Answer 1

• Convexity of preferences (and hence convex indifference curves) requires that for any two bundles A and B on the same indifference curve, every convex combination (every “mixture”) λA + (1 − λ)B is at least as good as A and B (for strictly convex preferences, strictly better for 0 < λ < 1). This is a global condition on all mixtures, not just on a few points.
• From the data we only know three discrete points are indifferent: (8,2), (5,3), (3,4). Convexity would require, for example, that the midpoint between (8,2) and (3,4), which is (5.5,3), is weakly preferred to (8,2) and (3,4). Nothing in the observation guarantees that. In fact you can construct preference relations (or utility representations) that make exactly those three bundles indifferent and every other bundle strictly worse – such preferences are non‑convex and produce a non‑convex indifference set.
• If, however, you additionally assume the usual regularity conditions used in indifference curve analysis (continuous preferences with diminishing marginal rate of substitution), then indifference curves are convex (they “bow in” toward the origin). But those are extra assumptions beyond the three observed indifferent bundles.