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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 ${\displaystyle (y,y)}$ is -1.
• The slope of the indifference curves at ${\displaystyle (2y,y)}$ is -1/8.

One such function is ${\displaystyle u(x,y)={\frac {1}{(x+ay)^{-3}+(ax+y)^{-3}}}}$, where ${\displaystyle a}$ 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.

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## References

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