# Throw-away paradox with independent goods

The throw-away paradox is a situation in which a trader can gain by throwing away some of his initial endowment.

The specific example, brought by Aumann and Peleg in 1974, concerns an economy with two commodities and two traders:

• In one situation, the initial endowments are (20,0) and (0,10). In competitive equilibrium, the bundle of trader A is (4,2).
• In the second situation, trader A throws away 10 units of commodity x, so the initial endowment is now (10,0) and (0,10). In equilibrium, the bundle of trader A is (5,5) - more of ''every'' commodity than in the first situation!

In their example, both traders have the same utility function, which has the following characteristics:

• It is homothetic;
• The slope of the indifference curves at (y,y) is -1;
• The slope of the indifference curves at (2y,y) is -1/8.

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

But, in this example, the products are dependent.

MY QUESTION IS: Is there an example of such a "throw away paradox", when the products are independent?

NOTE: both products are goods (have non-negative utility).

• What do you mean by "(in)dependent"? Linearly separable? Oct 6 '15 at 18:11
• @HerrK. I mean that they are not substitute goods nor complementary goods. I think this means that the utility function can be written as a sum: $u(x,y)=u_x(x)+u_y(y)$. Is this correct? Oct 7 '15 at 7:13

## 2 Answers

It seems (from the first page, at least; I don't have JSTOR access right now) that the essence of Aumann and Peleg's example is that

this rise in price of the commodity [trader A] holds is more than enough to compensate for the drop in the amount.

One similar situation from history occurred during the New Deal, when farmers were paid to plough over cotton fields and send hogs to be slaughtered for no reason, in order to take excess quantities of these goods off the market.

Obviously, the difference here is that this was a centralized intervention in a decentralized (and huge) market. The set of incentives are more complicated, and the many "trader A's" in this scenario were compensated for throwing away their goods. But the sentiment is very similar.

• It often happens, even today, that farmers decide to throw away some crops in order to raise the price. But, I am still looking for a particular example in which the involved goods are independent (or, a proof that this paradox cannot happen when the goods are independent). Oct 8 '15 at 7:48

This might be a trivial case, but what if the utility from one of the goods is negative? That would result in utility being gained by throwing away the good, but it would also eliminate any trading.

• You are right, I should have added a monotonicity assumption. Oct 8 '15 at 7:44