Prices are central to economic co-ordination in the neo-classical market model. Standard economic theory makes a good case for market (i.e. price and quantity) determination when there are many agents on both the demand and supply side. This process is assumed to be flawless and efficient. The theory is not nearly as sufficient in the case of bilateral monopoly bargaining, where just one buyer and one seller must decide between them the terms of trade. The division of the core is the bargaining problem and gives rise to the indeterminacy in the theory.

This essay is an examination of the issues at stake in bargaining and of the contribution by modern literature to the solution of the indeterminacy. I shall begin by examining simultaneous bargaining (Edgeworth and Nash) and then move to sequential bargaining with Rubinstein. Following will be an analysis of the contingencies in bargaining models i.e. commitment, time, and information. Recent literature will be reviewed, and examples of current applications of bargaining in the field of industrial organisation presented.

model to which the agents would eventually end up after bargaining. He proposed that the two would trade to a Pareto optimal outcome that left each at least as well off as he would be in the absence of trade.

The problem with Edgeworths analysis is that the final solution to the terms of trade is indeterminate [10] One claim made is that the solution or outcome will depend on relative bargaining strengths. The question is, how do you model these psychological factors?

Trade will occur i.e. each player can potentially be made better off by moving to a point on the core. But, what parameters are causal in determining the outcome, ability to commit? Outside options if bargaining collapses? Patience? As we shall see, all these issues and more, are of importance. Already there is a sense that if we could be more precise about the protocols and institutional arrangements of bargaining then the solution would become more determinate.

The greatest development for bargaining theory came from Nash (1950, 1951, 1953).[11] Nashs intention with his axiomatic approach was to refine away the noise that surrounds bargaining to leave the bones of what actually occurs within all bargaining contexts. The desire was not to formulate the bargaining procedure but to identify common characteristics in bargaining and in its solutions. Nashs approach was successful and he claimed a unique outcome.

The Edgeworth and Nash analyses are almost equivalent. Nashs diagram is a reformulation of Edgeworths in a utility space rather than a quantity space. The endowment point, the Pareto frontier (contract curve), the set of feasible bundles, and the core of the model are common to both.

Nashs analysis is co-operative in flavour and based on fixed outcome bargaining. The initial assumptions are that X is the set of feasible payoffs, û is the disagreement point, and U* is the Nash bargaining solution. The axioms are:

1. Invariance: the solution is invariant to equivalent utility transformation (affine only) i.e. the solution is independent of the units in which we measure utility.

2. Efficiency: the solution is Pareto optimal i.e. at least one player is made better off.

3. Independence of Irrelevant Alternatives: if we drop some set of possible utility combinations from X leaving a smaller set Y, then if U* was not one of the points eliminated, U* does not change.

4. Anonymity (Symmetry): switching the labels on the axes does not alter the solution. In some sense this implies an equal bargaining power assumption.

Nash states that:

i.e. U* is that value or outcome which satisfies the maximisation problem as stated, subject to the underlying constraints. It is equivalent to a Lagrangean constrained optimisation problem.

Nashs theory implies a unique outcome, U*. The uniqueness of the solution is guaranteed by the convexity properties of X, much like the convexity conditions on preferences in the Edgeworth analysis.

It is possible to analyse changes in bargaining power in the Nash context by introducing a mathematical and diagrammatic approach that uses marginal utility reinterpreted as an interest in the good at the margin. A similar analysis can be evoked to show that risk aversion imposes a penalty on the bargainer - ones outcome diminishes. The risk aversion effect can be obtained by applying a transformation to the risk averse players utility function.

A more recent and considerable development in bargaining theory was by Rubinstein (1982). Whereas the Edgeworth and Nash models were static, the Rubinstein model introduces the concept of dynamism to bargaining. In addition to its dynamic nature, the Rubinstein model is built on non-co-operative game theory, i.e., where commitments are not binding. Game theory is interactive, rational economic decision-making, utilising mathematical and intuitive models to analyse instances of co-operation and conflict between economic agents. Bargaining requires exclusively dynamic extensive-form analysis rather than strategic normal-form analysis.

The Rubinstein model is as follows: Assume two players. One makes an offer. The other can either accept or reject. As a tie break rule assume that an indifferent offer is accepted. Utility is discounted. The discounting of payoffs implies delay is costly. Assume different discount factors for the two players, representative of different degrees of impatience. Assume time passes between offers and counter-offers.

The conclusion of the model is that players realise the others degree of impatience and make an optimal offer, taking account of relative impatience. It can be shown that if player 1 is to make the offer, the outcome is:

If player 2 is to make the offer

if the discount factors were equal

The outcome of the sequential, alternating offer/counteroffer game is immediate
and efficient. Rubinstein shows that the equilibrium is a subgame perfect
equilibrium i.e. a Nash equilibrium based on credible threats.^{ 12}

The outcome of the game exhibits the first mover advantage concept: he who moves first has the advantage in that he can determine what the other is allowed to take in the fixed output bargaining model. The follower will wish to avoid delay and so may accept an unreasonable offer. The eventual outcome will depend on relative discount factors, i.e. on to whom delay is more costly. The larger output will go to he who can remain unaffected by the passage of time.[13]

The Rubinstein model is susceptible to changes in bargaining protocols. From Kreps (1990) and Sutton (1986):

1. The existence of an outside option for either of the players does not affect the Rubinstein subgame perfect equilibrium. This may be an unintuitive result. You may expect an outside option to improve bargaining power.

2. If player 1s discount factor is less than 2s, then player1 has a better bargaining position as he is more patient.[14]

3. The quicker a player can respond with another offer the greater their share from fixed bargaining.

The deficiency of each is that the outcome is immediate and efficient. Bargaining is not delayed. Conflict does not occur. This is in contrast to what we observe in reality (See below). The main criticisms of Rubinstein concern its unrealistic solution (immediate and efficient) and its very specific bargaining procedure (perfect information alternating offer or counteroffer).

Rubinsteins analysis makes extensive use of game theory. There are numerous problems with games. How do you define a game? When is it completely presented? What is the appropriate equilibrium / solution technique?[15] It is left to the model builder to decide these options. He must distil from reality the necessary details of commitment, time and information upon which the solution of the model is contingent. For slight variations in the specifics, the institutional protocols, one will be defining a different game with a different solution. The problem is allowing enough of the relevant economic factors into the formal, mathematical game analysis. But, how do you model judgement and intuition?

The contention of the Nash Program[16] and the course of action recommended by Sutton (1986) to further develop bargaining, is the examination of contingent-specific models, i.e. analyse the effect of time, commitment, and information on the outcome of bargaining problems. The hope is to refine Nashs bargaining axioms. Time, in the form of delay, is costly and ill be avoided by rational players[17]. Commitment refers to the ability to commit to bargained over outcomes. It represents trust[18]. Information is an asymmetrically distributed scarce resource. Unevenly distributed information is a cause of anomalies (e.g. conflict) in the models. There is the dual problem of imperfect and incomplete information[19].

Lyons and Varoufakes (1989) introduce a reputation argument to explain the occurrence of conflict. One would expect rational and well-informed agents to avoid costly delay, but unions, by delaying agreement can in some sense signal a tougher reputation in bargaining through willingness to accept the costly delay. Conflict is thus exhibited by relaxing the perfect and complete information assumptions [20]. This is where the literature has proliferated. See Crawford (1982).

Cramton (1984) examines issues in the urgency of bargaining in the incomplete information setting. There is the need to acquire information, which is achieved through analysing your opponents bids. Waiting is costly, so there is an incentive to optimally communicate to resolve bargaining. Cramton shows that the outcome depends on whose information set is restricted. There may be an incentive to lie or manipulate the situation with signal bids. The revelation principle does however rectify this[21].

Fudenberg and Tirole (1983) examine how changes in bargaining costs, the size of the contract zone, and the length of the bargaining process can effect aspects of the solution such as the probability of impasse and the likelihood of concession.

One can also consider the implications of the outcome for industrial organisations of the debate on bilateral monopoly determination. Is the Walrasian equilibrium of perfect competition robust to how we define the bargaining process between the agents in the model? Markets are assumed to be frictionless, and trade occurs flawlessly. What, if like Gale (1986), one assumes that agents matched randomly cannot agree on terms of trade thus incuring a search cost in looking for their next match? The indeterminacy of bargaining imposes a negative externality on the operation of a market with perfectly competitive characteristics.The Walrasian equilibrium no longer holds. Could vertical restraints affect bargaining positions and powers? Would the cost of delays from incomplete information be enough to encourage vertical integration to avoid these costs? For an analysis of the incentives of firms to merge because of the bargaining problem, see Horn and Wolinsky (1988). The bargaining problem may not be eliminated by vertical integration. In the vertically integrated firm the inputs of one division are the outputs of another. If management was decentralized then the interdivisional costs would be determined by bargaining. These are the transfer prices. These transfer prices are subject to influence activities and costs[22] indicating that bargaining can still impose a negative externality on firms, even after vertical integration to remove them.

Bargaining has been central to economic theory throughout its history. Its apparent indeterminacy has been its deficiency. The resolution of the (in)determinacy problem has been addressed in static and now dynamic-interactive frameworks. The literature has proliferated and is coalescing on a contingent-specific approach. The indeterminacy is now being appreciated as the of real-life anomalies such as the absence of Walrasian prices, conflict, mergers, transfer prices etc.. Indeterminate bargaining - a hindrance to theory and an indication of reality.

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