Question

4. We want to think about the role of convexity of preferences (along with strict monotonicity of preferences) in consumer choice (a). In the consumer's utility maximization problem, explain why if prefer- ences are strictly convex, the ordinary demand choices are unique. (b). Now, explain if they are just convex, the claim in part (a) is not neces- sarily true. (c). Per part (b), now explain why with convex preferences (also strictly monotone), the optimal set of demands at any fixed price and income forms a convex set. (d). In the consumer's expenditure minimization problem, if preferences are strictly convex and strictly monotone, explain why the optimal Hicksian demand is unique. (e). Now, in the consumer's expenditure minimization problem, if pref- erences are nonconvex, explain why the optimal solution are not necessarily unique. (f). In part (e), are do the optimal demand choices at a fixed price and income form a convex set?

Answer 1

e Fisst of all convexity of preferences Teams am novi dual customer's ordesing of voious outcomes with the propesty 9 that that "Averages are "Averages aude better than that of extremes". (a) Im the consumed 's unity maximization problem the main thing what carasume umts is to maximize the utility with given prices income level. AC preferences ane Stoictly convex, there wi be only one optimal choice on ench budget the because in this type of preferences congumer chooses bundle 1 O pee fess more likely bundle 1 over and ) bundle 2 of goods. This is why, oodi nary Demond choices are unioue when preferencesc aure Stofctly convex a) Psefesences are strictly conver, and moncone that is due to a bundle 'z' is preferred over 'p' lie z> when both arse affordable. As Hickbian demand shows or describes the sention between poice and quantity demanded of a good suming price of other goals f level of utility semaitng congtante >Therefore, HRkpian optimal demand if unique pocage of strict convex f monotonic preferences. @). In nhimization pocblem if preferences are non-convex optimal solution are met neceseosily unique because there will be more than one choice on eauch budget line → Hence, optimal solution is not at all unique. (f). If preferences are non-convex, then the optimal demand choices does not form a convex pet at a fixed Price and income level as it does not fatisfy tangency condition. (b) In case of convex preferences, these won't be only me optimal choice on each budget line, this is because cony point that satisfies the tengency condition must be an optimal point. Thesefore claim a" is not necessarily toue in this coupe. (c) preferences are conver and Stohtly monotone indicate mose 7 better till before optimal point. The optimal demands at any given prices and income level fooms convex get because any set of demand that gatis files tangency condition ie (Indifference clove if tangent to the budget time will be optimal and also les entirely in the get the repore in order to reach at optimal point there will be few demands with given prices and income.