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## August 21, 2004

### No-Trade Theorems; L. Samuelson(2004)

I was reading Larry Samuelson's survey, "Modeling Knowledge in Economic Analysis," in the June 2004*Journal of Economic Literature*. Much of it is about No-Trade Theorems. I'll modify one of his first examples to illustrate.

Basic Model. Alice owns 1 share of a company. That share is worth $200 if the company's new product will be a success, and $300 if it is a failure. Each of these has equal probability, so the market price is $250.

Alice can make one take-it-or-leave-it offer to sell the stock to Bob. Clearly, so far Alice would offer P=250 and he would accept, but both players would be indifferent. We don't really have a model of trade yet, since a small transaction cost would block all trade.

We will think about adding two additional assumptions.

Assumption 1: Alice is better informed. Alice finds out whether the product will be a success, Bob knows she has found out, she knows Bob knows, and so forth (her finding out is "common knowledge", though whether she has found success or found failure is unknown to Bob).

Under Assumption 1, if Alice find out FAILURE, what happens? She will believe (correctly) that the value is $200, and Bob will believe it is $250. But now if she offers to sell to him at P=250, he will change his belief to value V=200 and refuse to buy. Indeed, the only equilibrium with trade is if Alice offers P=200, and, again, both players are then indifferent about trade.

This is a No-Trade result. Our intuition that difference of opinion will result in the low-valuer Alice selling to the high-valuer Bob fails, because the very act of Alice trying to sell converts Bob to being a low-valuer.

Most of Samuelson's survey looks at papers that generalize the result to more complicated situations than this little example and to fancy ways to try to generate trade, many of them fiddling with the standard Bayesian assumption of common priors (that both players know the probability of failure is .5, that both of them know Alice has found the information, etc.) But I wonder whether the paradox can be resolved even within this little example.

Suppose if instead of Assumption 1, we used Assumption 2.

Assumption 2. With probability .1, Alice gets into a fight with the president of the company, and holds a grudge which means her share of stock is worth $95 less to her than to anyone else in the world. Bob does not know whether she really had the fight, but he knows the probability is .1 and the consequence is a $95 difference.

Under Assumption 2, if Alice has the fight, then she will offer P=250 to Bob and Bob will accept. Unlike in the basic game, Alice now has a strong incentive to sell-- she is not indifferent. Bob is still indifferent, but that is an example of the purely technical "open-set" problem-- Alice would be willing to offer P=249 if she had to, and Bob would then be strongly desirous of accepting.

Assumption 2 is an example of a non-informational reason for trade, a reason that requires trade to attain efficient allocation of resources. This is, of course, the second reason we intuit for why trade occurs. It is by far the main reason for trade in goods, and it is also important for trade in securities, though Samuelson and others argue that efficiency reasons can't explain the volume of trade in securities.

Now let's use both Assumption 1 and Assumption 2. Note that this means that with 100% probability Alice has an informational reason for trade, but with 10% probability she also has an efficiency reason. What will happen?

First, suppose Alice hears that the product will be a failure. She will offer to sell to Bob at some price P*. She will tell Bob that she is selling because she had a fight with the president, but Bob won't believe that. He knows that with high probability she is selling because the company's value is only 200. What is the highest value of P* that Bob will accept?

If Alice had no fight and heard that V=300, she would offer P=300 (or make no offer) and Bob would deduce what happened. This has probability .9 (.5).

With probability .9(.5), Alice had no fight but heard that V=200 and is selling for that reason.

With probability .1(.5) Alice had a fight and heard V=200, and so has two reason to sell.

With probability .1(.5), Alice had a fight and heard V=300, and so will sell if P*>205.

That means that if P*>240, a sale will occur with probability .45+.05+.05= .55.

Bob's expected payoff from accepting P* is zero if

[(.1)/ (.55)] [.5 (300) + .5 (200) - P*] + [.45/ (.55)] [200-P*] =0.

This reduces to

(2/11) (250-P*) + (9/11) (200-P*)=0,

500 -2P* +1800 - 9P* =0

2300 = 11P*

P* = 2300/11= 209 (approximately)

If P* =209, then Alice is willing to sell even if she heard good news, if she really had a fight with the president, and Bob is willing to accept her offer, because he can at least break even (and if Alice offered 208, Bob would be strongly willing to accept).

Thus, a small probability of an efficiency reason for trade has generated a high probability of trade. Trade will occur 55% of the time, but 45/55 of the time trade occurs, its direct motivation will be Alice's superior information, not her possible efficiency motivation. So if you think that most securities trading is not motivated by efficiency, this model explains what is going on. The market (Bob) knows that most people are selling because they have private information, but the market is only willing to trade with them because it knows that some people are selling for efficiency reasons.

And I've shown what is going on with a much simpler and more conventional model than what's in the literature. To be sure, it's just a numerical example, but it's got 90% of what we need for an explanation of the real-world factoid.

Still, it might be worth expanding a bit, if this is not already in the literature. It would be interesting to see what would happen if Alice's probability of being informed is not 100%, but X%, and compare the effect of X with the effect of the probability of having an efficiency reason (here, 10%).

Posted by erasmuse at August 21, 2004 11:49 PM

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