Throw away paradox explained

In economics, the throw away paradox is a situation in which a person can gain by throwing away some of his property. It was first described by Robert J. Aumann and B. Peleg^[1] as a note on a similar paradox by David Gale.^[2]

Description

There is an economy with two commodities (x and y) and two traders (e.g. Alice and Bob).

• In one situation, the initial endowments are (20,0) and (0,10), i.e, Alice has twenty units of commodity x and Bob has ten units of commodity y. Then, the market opens for trade. In equilibrium, Alice's bundle is (4,2), i.e, she has four units of x and two units of y.
• In the second situation, Alice decides to discard half of her initial endowment - she throws away 10 units of commodity x. Then, the market opens for trade. In equilibrium, Alice's bundle is (5,5) - she has more of every commodity than in the first situation.

Details

The paradox happens in the following situation. Both traders have the same utility function with the following characteristics:

• It is a homothetic utility function.
• The slope of the indifference curves at

(y,y) is -1.

• The slope of the indifference curves at

( 2 y,y) is -1/8.

One such function is

u(x,y) = (1)/((x + ay) ^{-3} + (ax + y) ^{-3}) , where a is a certain parameter between 0 and 1, but many other such functions exist.

The explanation for the paradox is that when the quantity of x decreases, its price increases, and the increase in price is more than sufficient to compensate Alice for the decrease in quantity.

See also

• Paradox of Plenty
• Resource monotonicity

Notes and References

1. 10.1016/0304-4068(74)90012-3. A note on Gale's example. Journal of Mathematical Economics. 1. 2. 209. 1974. Aumann. R.J.. Peleg. B..
2. 10.1016/0304-4068(74)90036-6. Exchange equilibrium and coalitions. Journal of Mathematical Economics. 1. 63–66. 1974. Gale. David.