Suppose the consumer has convex preferences, and we know (2, 4) ~(6, 1). Is
the statement (4,2.5) >=(6, 1) correct? What about the statement (4, 2.5)>
(6, 1)? Suppose now the consumer has strictly convex preferences, and and
we know (2, 4) ~ (6,1). Does any of your answers change?
no
nio
To determine the correctness of the statements (4, 2.5) >= (6, 1) and (4, 2.5) > (6, 1), we need to understand the concept of preferences and their relation to convexity. Let's start with the first part of the question about convex preferences:
Convex preferences imply that consumers prefer a combination of goods that lies on or inside a convex set. In other words, if (a, b) and (c, d) are two bundles of goods, and combination (a, b) is preferred to (c, d), then any combination lying on the line segment connecting (a, b) and (c, d) will also be preferred to (c, d).
Given that (2, 4) ~(6, 1), the tilde (~) notation represents indifference between two bundles, meaning the consumer has no preference between them. It implies that the consumer is equally satisfied with both (2, 4) and (6, 1).
Now, let's evaluate the statements based on these preferences:
1. (4, 2.5) >= (6, 1):
This statement compares bundle (4, 2.5) with (6, 1). Since (4, 2.5) is not preferred to (6, 1) (as the consumer is indifferent between (2, 4) and (6, 1)), this statement is not correct. The inequality sign (>=) implies a preference relation, which is not satisfied in this case.
2. (4, 2.5) > (6, 1):
Similarly, this statement compares bundle (4, 2.5) with (6, 1). As the consumer is indifferent between (2, 4) and (6, 1), this statement is also not correct. The inequality sign (>) implies a strict preference relation, which is not satisfied in this case.
Now, let's move on to the second part of the question, where the consumer has strictly convex preferences:
Strictly convex preferences mean that consumers strictly prefer combinations that lie inside the convex set. In this case, even small movements along the line connecting two bundles will result in a preference for the bundle closer to the midpoint of the line segment.
Considering that (2, 4) ~(6, 1), the fact that the consumer has strictly convex preferences does not change the evaluation of the statements.
Therefore, the answers remain unchanged. Both statements, (4, 2.5) >= (6, 1) and (4, 2.5) > (6, 1), are still not correct based on the given information, regardless of whether the consumer has convex or strictly convex preferences.