Question

1. In a two-good world, let - bundle x 3 loaves of bread and 0 apples; - bundle y 0 loaves of bread and 10 apples; - bundle z 2 loaves of bread and 4 apples. Suppose a consumer weakly prefers x to y, weakly prefers y to z, and strictly prefers z to x. (So, this consumer's preferences violate transitivity.) Explain how you can extract money from this person a. b. Do this consumer's preferences over x, y and z violate monotonicity? Why or why not?

Answer 1

a.

Say the consumer starts off with bundle x. Since he strictly prefers z to x, he gets more value from consuming z than from consuming x. Assume the consumer gets $1 worth of benefit from consuming z over x (you can pick any other positive number instead of $1, and the reasoning will still work). So, he is willing to give up x and pay an additional $1 to consume z. Now, say, you offer to sell him z in exchange for x, and along with that he has to pay you $0.5. He will happily agree to. And you end up with $0.5.

Now, the consumer weakly prefers y to z. That means, he is will agree to trade y for z, because he likes y either as much as, or more than z. So you trade z for y (but don't take any extra money).

Next, since the consumer prefers x to y, you can offer to exchange his y for x, and he will agree (and again, you don't take any extra money). So he trades his y with you in exchange for x.

Notice, the consumer started off with x and ended with x. But in this processes, he lost $0.5. He only kept getting into exchange agreements with you because he thought he would be better off. But he's ended up back where he started, but now with $0.5 less.

But you can do this all over again. Again offer to sell z in exchange for x and $0.5. Once he agrees, exchange his z for y. And then exchange his y for x. And he would've lost another $0.5.

If you continue to do this, you can keep extracting money out of him.

b.

The consumer's preferences violate monotonicity if he prefers less bread (but the same amount of apples) to more bread, or less apples (but the same amount of bread) to more apples, or less of both bread and apples, to more of them.

You're told that the consumer strictly prefers z to x. This means that he prefers 2 bread and 4 apples to 3 bread and 0 apples. This doesn't violate monotonicty since one bundle has more of bread while another has more of apples. A useful rule of thumb is that monotonicity is violated if he prefers a bundle that has less of both than one that has more of both. This is not true here.

Next, you're told that he prefers y to z. That means he prefers 0 bread and 10 apples to 2 bread and 4 apples. Again, this preference does not violate monotonicity because one bundle has more of bread while another has more of apples.

Finally, he prefers x to y. That means he prefers 3 bread and 0 apples to 0 bread and 10 apples. Again, this doesn't violate monotonicty because one bundle has more bread while another has more apples.

So, the consumer's preferences do not violate monotonicity.