Saturday, 21 May 2011

Just for fun: the implementation criticism and the theory of the firm

As I have noted in a number of previous posts the incomplete contracts (or property rights) approach to the theory of the firm has become a workhorse in the literature. However it has been argued that there is a problem with the foundations of the theory of incomplete contracts that makes their use for the theory of the firm problematic. The standard property rights approach assumes there is symmetric but unverifiable information. That is, there is information that the contracting parties observe but a third party, e.g. the courts, cannot verify. Such information leads to incompleteness of contract since the inability of the courts to verify such information means it cannot be contracted on. What Aghion and Holden (2011) call the “implementation criticism” basically says that observable information can be made verifiable by the use of cleverly designed revelation mechanisms.

Aghion and Holden (2011) give a very nice toy example of how such mechanisms work:
Here, we begin with an example of a revelation mechanism that is drawn from Aghion, Fudenberg, Holden, Kunimoto, and Tercieux (2010). There are two parties, a buyer B and a seller S of a single unit of an indivisible good. If trade occurs, then B’s payoff is V_B = theta – p; where theta is the value of the good to the buyer and p is the price. S’s payoff is just V_S = p. The good can be of either high or low quality. If it is of high quality, then B values it at 14; if it is of low quality, then B values it at 10, thus theta is in {10, 14}.

Suppose that the quality theta representing the true value of the good to the buyer is observable and common knowledge to both parties. Even though theta is not verifiable by a court, and therefore no initial contract between the two parties can be made credibly contingent upon theta, truthful revelation of theta by the buyer B can still be achieved through the following mechanism:
1. B announces theta to be either “high” or “low.” If he announces “high,” then B pays S a price equal to 14 and the game then stops.
2. If B announces “low” and S does not “challenge” B’s announcement, then B pays a price equal to 10 and the game stops.
3. If S challenges B’s announcement then:
a) B pays a fine F to T (a third party), and
b) B is offered the good for 6.
c) If B accepts the good, then S receives F from T (and also the 6 from B) and we stop.
d) If B rejects at stage 3b, then S pays F to T, and
e) B and S Nash bargain 50:50 over the good.
When the true value of the good is common knowledge between B and S, this mechanism yields truth telling as the unique (subgame perfect) equilibrium. To see this, let the true valuation be 14; and let F = 9: If B announces “high,” then B pays 14 and we stop. If, however, B announces “low,” then S will challenge because, at stage 3a, B pays 9 to T and, this cost being sunk, B will still accept the good for 6 at stage 3b (because it is worth 14 and 14 – 6 = 8 is greater than 14/2 = 7, which is what B gets if it rejects the offer of 6). Anticipating this, S knows that by challenging B, S receives 9 + 6 = 15, which is greater than the 10 that S would receive if S did not challenge.

Moving back to stage 1, if B lies and announces theta = 10 when the true state is theta = 14, B gets 14 – 9 – 6 = –1, whereas B gets 14 – 14 = 0 if B tells the truth.
One obvious question this gives rise to is, If such mechanisms can get around incompleteness of contract, why are contracts still incomplete? Or, Why don’t we see such mechanisms being used?

One reason may well be that these mechanisms are not robust to even small deviations from common knowledge. Strict common knowledge is a very strong assumption which is very unlikely to hold in the real world. What it requires is that if party A knows something, party B knows it, party A knows that party B knows it, and so on ad infinitum, all with perfect certainty. The problem with this is that it has been shown that the type of mechanism used above may not work if there is only "close to common knowledge": that is, party A knows something, party B knows it, party A knows that party B knows it, and so on ad infinitum, but in each case only with almost perfect certainty. What has been shown is that if any mechanism can achieve truthful revelation as an equilibrium under common knowledge, then under approximate common knowledge, there must also exist an equilibrium with nontruthful revelation. Or in other words, the mechanisms discussed above turn out to be fragile in the sense that they depend crucially on delicate assumptions about higher-order beliefs.

Another possible reason for not seeing the above type of mechanism in the real world is the effects of asymmetric information. An assumption in the mechanism outlined above is that there is symmetric information between the contracting parties. That is, the both B and S know theta. It is the third party, the courts, who do not know what value theta takes. The above mechanism breaks down with the introduction of only small amounts of private information. For example, if S does not know the quality of the good then he cannot challenge B’s announcement of the quality.

Aghion and Holden make the point that
[ ... ] an interesting direction for future research is to explore how an incomplete contracts/property rights model is affected by aspects of common knowledge, revelation mechanisms, and bargaining under asymmetric information. It seems plausible that these issues could lead to other reasons for inefficiency in investment, and property rights in the form of asset ownership may help to alleviate these inefficiencies.
  • Aghion, Philippe, Drew Fudenberg, Richard, Holden, Takashi Kunimoto, and Olivier Tercieux (2010). 'Subgame-Perfect Implementation under Value Perturbations and the Hold-Up Problem', Unpublished working paper.
  • Aghion, Philippe and Richard Holden (2011). 'Incomplete Contracts and the Theory of the Firm: What Have We Learned over the Past 25 Years?', Journal of Economic Perspectives, 25(2) Spring: 181-97.

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