Eric Rasmusen
21 November 2006

Jean Tirole has a nice example illustrating the hold-up problem and authority in his I.O. book. Here is the game, modified somewhat.

0. The buyer, a retail chain owner, is considering buying soda pop from the seller, who owns a factory that produces it. The buyer's laboratory is at work on a new chemical additive that he plans to call Zantac. It will cost the seller amount C in extra production costs to add the new chemical to the drink, if it is indeed added. But Zantac might be a failure, not worth adding given its production cost.

1. The buyer and seller agree to a contract specifying who has the authority to decide whether Zantac is added. 2. The buyer chooses a probability X that Zantac will taste delicious. To get probability X costs the buyer I = X^2/2 in research costs. If Zantac tastes delicious, it will add V>C to the buyer's value of the soft drink. If Zantac tastes less good, it will add B<C, where B might be positive, zero, or negative, depending on our scenario. Start with B>0, even though this is not Tirole's case.

3. After Zantac's value is discovered, that value is known to buyer and seller, but not to anyone else, so it can't be contracted on.

4a. Buyer and seller bargain over whether Zantac will be added and whether the buyer will pay some amount P to the seller (where P could be negative).

4b. If buyer and seller can't agree, whoever has the authority makes the decision.

We assume that the courts can observe whether Zantac is added to the soft drink, but not whether Zantac is delicious or not. Thus, the players can make a contract to add Zantac, but not a contract to add Zantac only if it is delicious.

What level of investment, X, will be chosen?

Let's start with the first-best. In the first-best, Zantac is added only if it is delicious, since V>C and B<C. Thus, the combined surplus of buyer and seller is

-X^2/2 + + (1-X)(0) + X(V-C)

Maximizing with respect to X gives the first order condition -X +V-C=0, so

X* = V-C (first best)

We will start with what happens if neither player has been granted the authority to decide whether Zantac will be added. Then both of them must agree in step 4a, or Zantac cannot be added.

Let's look at the influence of bargaining power in cases 1 to 3.

1. Suppose the seller has all the bargaining power , meaning that he can offer the buyer a price P for adding Zantac and the buyer just accepts or rejects.

The seller will offer price P=V, and the buyer will accept. The buyer gets none of the surplus from Zantac being delicious. As a result, the buyer's overall payoff is -X^2/2 + X(V-V) and he will choose X=0-- no research.

A solution would be for the buyer to sell the laboratory to the seller. Then the seller would choose X efficiently, because he gets 100% of the marginal benefit. That is one big lesson of these models: who owns particular assets such as a laboratory can be important for efficiency.

2. Suppose the buyer has all the bargaining power, meaning that he can offer the seller a price P for adding Zantac and the seller just accepts or rejects.

The buyer will offer the price P=C and the seller will accept (the seller would reject any lower price). The buyer's overall payoff is -X^2/2 + X(V-C). This is the same as the combined surplus we used in the first-best, because the buyer gets all the surplus. Thus, the buyer will choose the first-best level of X, X= V-C.

Case 2 illustrates another principle: the problem arises because of market power. One situation in which the buyer has all the bargaining power is if there are many sellers, all with the same production cost C, and the buyer is free to choose among them. In that case, the investment I is not relationship- specific--- and spot-market transactions do just fine in maximizing social surplus.

3. Suppose the buyer and seller have equal bargaining power and split the surplus 50-50. This is the case we will explore in detail when we come to look at authority being granted to one party or the other.

Let's work back from the end of the game. If Zantac is a failure, there are no social gains from adding it, so the two players will not come to an agreement to do so. If Zantac is a success, adding it creates social surplus of V-C. With equal bargaining power, the two players split this, each of them getting (V-C)/2 of it. They will agree that the buyer will pay the seller the price P=(V+C)/2. This will give the buyer V-P = V- (V+C)/2 = (V-C)/2 of the surplus, and the seller will get P-C = (V+C)/2 -C = (V-C)/2 of the surplus.

Foreseeing this, the buyer must decide how much to invest in research, what level of X to pick. His expected payoff is

-X^2/2 + (1-X)(0) + X(V-C)/2

The first order condition is -X + (V-C)/2 = 0, so

X = (V-C)/2 = V - (V+C)/2

This is below the first-best level of X*= V-C, so there is underinvestment. That happens because the buyer knows he will only end with half the surplus that investment creates.

Cases 1, 2, and 3 were what happens when neither party is given authority, so each player is a "veto gate", able to block Zantac from being added.

All three cases depended on a background legal rule that we take for granted: that if the players cannot agree, Zantac is not required as a feature of the soft drink. We could imagine another legal regime in which if trading partners cannot agree on product quality they must trade the highest possible quality, without any change in the agreed-upon price. Then the threat point would be to add Zantac with P=0 if the players could not come to an agreement, whether (since B>0 and Zantac does increase quality) Zantac was a failure or a success.

The next cases will explore what happens if the players start with a contract that gives one or the other of them the authority to require Zantac to be in the soft drink.

4. Suppose the seller has the authority to add Zantac or not, without any need for agreement from the buyer. (We will keep the 50-50 bargaining power assumption for the rest of these notes.)

If the game reaches move 4b, the seller will not add Zantac, because it costs him C to add it. Thus, in move 4a the seller has a credible threat to not add Zantac unless the buyer pays him some extra money. How will the surplus of (V-C) be split? Exactly the same as in Case 3: (V-C)/2 for each player. As a result, the buyer's investment incentives are the same as in Case 3.

X = (V-C)/2

Again there is underinvestment. The reason things work out the same as in case 3 is that the buyer has no motive to veto Zantac. So long as the seller can't raise the price without the buyer's consent, the buyer has no reason to block a quality increase. It is the seller's consent that matters, and this would be true even if C=0, since he would not sell his consent, a valuable item, for free.

5. Now let the buyer have the authority to require Zantac.

If no new agreement was reached in move 4a, then in move 4b the buyer would require Zantac, because Zantac would increase the quality. If Zantac is a success, it adds V to the buyer's payoff. If Zantac is a failure, it still adds B>0 to the buyer's payoff.

Looking ahead to this, what will happen in move 4a when they bargain? If Zantac is a success, the buyer has no reason to agree to pay anything to the seller, since he knows he can add Zantac by waiting till move 4b and exerting his authority. But if Zantac is a failure, something interesting happens. The social surplus from adding Zantac is negative, because it is B-C, the difference between its value to the buyer and its cost to the seller, and B<C. Thus, there are social gains if they can agree not to add Zantac instead of proceeding to move 4b and adding it. They will split these gains, so each of them will get (C-B)/2 of the surplus. This happens when P = -(C+B)/2, which means that the seller pays the buyer not to require Zantac to be produced.

The buyer's overall payoff is thus, if we denote the payments from buyer to seller in the cases of failure and success as Pf and Ps,

-X^2/2 + (1-X) [Pf] + X(V-Ps) = -X^2/2 + (1-X)(C+B)/2 + X(V-0)

This has first-order condition -X -(C+B)/2 + V = 0, so

X = V - (C+B)/2

This is above the first-best level of X*= V-C, since (C+B)/2<(2C)/2, so there is overinvestment. The buyer's return from his investment exceeds the social return, because if Zantac is a success, he gets V instead of V-C. Moderating this excessive return to successful investment is the fact that the buyer's return from unsuccessful investment has also risen. If Zantac is a failure, he can at least extort (C+B)/2 from the seller in exchange for not requiring inefficient production. The overall effect, though is that investment is too high

Think back to cases 1 to 3, where we looked at bargaining power. Giving all the power to the investing player, the buyer, reached the first-best. With authority it is not true that we get the first-best just by allocating the bargaining power to the correct player. We get inefficiency even in case 5, where we give the authority to the investing player.

What can we do?

6. Agree in advance to roll dice to determine authority, after the buyer has made his investment of I=X^2/2

What probability should be used? If the probability the seller gets the authority is Z, the buyer's expected payoff, viewed from the time when he must choose X, is a weighted average of cases 4 and 5:

Z[ -X^2/2 + (1-X)(0) + X(V-C)/2] + [1-Z] [-X^2/2 + (1-X)(C+B)/2] + X(V-0)]

This has the first order condition

-X + Z(V-C)/2 - [1-Z](C+B)/2 + [1-Z]V =0

X = ZV/2 - ZC/2 - C/2 - B/2 + ZC/2 + ZB/2 +V -ZV = [V - C] + C/2 - ZV/2 - B/2 + ZB/2

To get the first best of X=X*=V-C we need

C/2 - ZV/2 - B/2 + ZB/2=0,

so C-B = Z(V-B )

and Z = (C-B)/(V-B) That gives us a value for Z that is between zero and one, since V >C>B.

The right authority probability (not 50-50) will give the buyer the authority with just enough probability that his expected payoff from investment is X(V-C) and he will choose X=X*. We know that because if the buyer gets the authority there is overinvestment and if the seller does there is underinvestment, but whoever has the authority, an efficient solution is reached regarding whether to add Zantac, once it has been researched.

There is another way, less peculiar, to get the first best:

7. Use a buyer-option contract. The price is set at P=C in advance, and the buyer has the authority to decide whether to add Zantac.

Work back from the end as usual. In move 4b, the buyer will use his authority to require Zantac only if it is a success, since V>C but B < C. This is efficient, so no bargaining need occur at move 4a. The buyer gets a surplus of (V-C) if Zantac is delicious and zero otherwise, so he capture the entire gain from his research, and he will choose X= X*.

It is not necessary for the courts to know whether Zantac is delicious or not for the buyer-option contract to work. All the court needs to observe is whether Zantac is added and whether the buyer pays the amount P=C to the seller. In fact, using the buyer-option contract it is not even necessary for the seller to observe whether Zantac is delicious, since the buyer makes all the decisions. That's different from the earlier versions, where the seller had to know whether the buyer's value was B or V so they could bargain over the surplus. If the seller didn't know the buyer's value, that would introduce a different inefficiency: that sometimes bargaining would fail and the decision whether to add Zantac would not always be efficient.

You may think: "Well, that's obvious. Why go through the first six cases when we can solve the problem just by not neglecting to specify the price in advance?" That question makes some sense in our model. I'll speculate as to an answer, but I haven't worked it out fully.

Suppose the players did not know the production cost C in advance, and only learned it after the buyer's research has been finished. Then, they would not know what price to specify in the buyer-option contract. They could pick it to be the expected value of C, but the problem with that is that then the price wouldn't necessarily induce the efficient decision in move 4b.

Suppose, for example, that the production cost might be either our old value of C, between B and V; or some higher cost equal to V+D. Let the probabilities of these two cost levels be independent of whether Zantac's value is B or V, and let the probabilities be such that the expected value E of the cost is less than V. A buyer-option contract with P=E won't induce efficient decisions in move 4b, and won't make the buyer's expected gain from investment equal to the expected social surplus. If the cost turns out to be C, the buyer will not get all the surplus from his investment, because the production cost is less than the price he pays. If the cost turns out to be V+D, the buyer will choose to add a successful Zantac in move 4b even though Zantac's value is less than its production cost. Thus, we would get back to the extortion problem in move 4a. Perhaps some other price than E would get the incentives correct, but I have a feeling it wouldn't. (This feeling could be checked out fairly easily; I just don't have time.)

Next, let's explore different values of B. Let us assume 50-50 bargaining power and that the buyer has the authority.

8. Buyer Authority, 50-50 bargaining split, B=0. This is the case in Tirole's book.

There are two equilibria. In one of them, if Zantac is a failure, then in move 4b the Buyer will use his authority to require Zantac. Thus, in move 4a the Buyer can extort the payment of (C-B)/2=C/2 from the seller in exchange for not making the seller produce Zantac. The buyer's overall payoff will be -X^2/2 + (1-X)(C/2) + X V and there will be overinvestment in much the same way as in case 5.

In the second equilibrium, if Zantac is a failure, then in move 4b the Buyer will use his authority to not require Zantac, since he is indifferent about requiring it. Thus, in move 4a the Buyer cannot extort the payment of (C-B)/2 from the seller. The buyer's overall payoff will be -X^2/2 + X V and there will be overinvestment, but not in the same way as in case 5. Instead, the investment size will be the same as in case 9, which follows.

9. Buyer Authority, 50-50 bargaining split, B<0. Zantac actually makes the drink taste bad.

Now if Zantac is a failure, then in move 4b the Buyer will definitely use his authority to not require Zantac, since he absolutely dislikes it. Thus, in move 4a the Buyer cannot extort the payment of C/2 from the seller. He can threaten to require Zantac if the seller refuses to pay him C/2, but the threat is not credible, because the seller knows the buyer won't want to make the product worse. On the other hand, if Zantac is delicious, the buyer will require it. The buyer's overall payoff will be

-X^2/2+ X V

The first-order condition is -X +V=0, so

X= V.

There will be overinvestment, and more than in case 5, where B>0 and X = V - (C+B)/2. The reason is that now the buyer does not have a credible threat and cannot extort from the seller. When the buyer could engage in extortion, it was only when Zantac was a failure, so the extortion reduces the difference between the buyer's payoff when Zantac was a failure and when it was a success, which reduces the buyer's incentive to invest more and choose a large X.

10. Seller Authority, 50-50 bargaining split, B<0 . Again, Zantac actually makes the drink taste bad.

This will work out just like case 4, when the seller had authority but B>0. At move 4b, the seller will not want to require Zantac, since it costs him C to produce. At move 4a, buyer and seller will agree to require Zantac only if it is delicious, and will split the surplus.

The reason to bring up this case is that the seller might be tempted to make a non-credible threat. If Zantac were a failure, he might say in move 4a, "Buyer, pay me B, or I will make the product worse by adding Zantac to it." The buyer's rational response is, "No, seller, you won't do that, because it will cost you C to add it."

These notes take off from the example on pages 31 to 33 of Tirole's book. They extend it by discussing bargaining power (which Tirole discusses earlier in his book) and the value of B, the failure value of the innovation. My claim that if B<0, so the innovation actually would hurt the buyer, then the buyer could not credibly use it as a threat against the seller, might be controversial. More can found on that issue in Lyon and Rasmusen (2004). The second half of Gibbons 2005 survey is a good place to read about more models in this style.

Robert Gibbons (2005) "Incentives between Firms (and within)" Management Science, 51,Issue: 1: 2-17 January 2005).

Thomas P. Lyon and Eric Rasmusen (2004) ``Buyer-Option Contracts, Renegotiation, and the Hold-Up Problem,'' (with ) Journal of Law, Economics and Organization, 20,1: 148-169 (April 2004).

Jean Tirole (1988), The Theory of Industrial Organization, Cambridge: MIT Press, 1988.

I thank Hubert Pun and Abhijit Ramalingam for their comments.