3P\ Dijj dnf
Definition 6. 1 (Bargaining Set) For given FI* = (Ilf ,11*) let
9? = {(n1,n2)|nI <
kx
Then, 9t is a collection of all payoff combinations that are individually rational, and over which the players may bargain. 9i is defined as the bargaining set.
Note that the Pareto efficient boundary of the bargaining set is the segment of the rent transformation frontier ( see equation 4.6') that has individually rational points. It has already been shown that if the elasticities of factor substitution in both the sectors are at least unity then the boundary of the bargaining set is concave to the origin
(Proposition 4.2, chapter 4). Therefore, the points in OCD form a convex set. Since the maximum output that a sector can produce in the event of specialization is finite
(because of finite endowment of mobile factors) and concave production function), and since also the rental income is always less than the level of output, the set 9^ is also bounded. The set includes its boundary points, hence it is closed. The set 9f is also a subset of a 2-dimensional Euclidean space, therefore it is compact and convex.
If the elasticities of factor substitution are sufficiently low to undermine the convexity of the bargaining set defined over certain outcomes, then as in the standard
case (for example Nash, 1950) we can invoke the expected payoff approach that will guarantee the convexity of the corresponding bargaining set. Any concave function defined over 9$ will have its maximum in it.
If the disagreement payoff n* is assumed to be predetermined and fixed, then the underlying bargaining problem of the tariff game satisfies the conditions of the existence theorem proved by Nash (1953). In the case that the disagreement payoff is not known a priori, then the bargaining problem in a tariff game satisfies the conditions of the existence theorem proved in Harsanyi (1963).
Thus, whether the disagreement payoffs are assumed to be predetermined or not (variable), the tariff game in cooperative form can be viewed2 as a standard bargaining problem. However, in this chapter we will assume that the disagreement payoff is predetermined3 and therefore, the bargaining problem is defined by the pair(SK,n‘i). Let B be the class of such 2-person bargaining games for which the underlying noncooperative game is given by T (see Chapter 5).
6.2 Nash Solution to the Bargaining Problem in a Tariff Game
Several solution concepts have been proposed in the literature. The Nash solution is one of them. A short review of these solution concepts can be found in Datt (1989). He has also convincingly argued that Nash solution has a degree of generality not shared by other solution concepts. Several other studies that attempted to solve standard bargaining problems by introducing various frictions into it have concluded that Nash solution is robust (see, Binmore, Rubinstein and Wolinsky, 1986; van Damme, 1986; Chun, 1988b; Chun and Thomson, 1990; Carlson, 1991).
2 It is very important to note that this study assumes that players are interested in maximization of the sectoral rents. How they use the rental income - whether they spend all in the consumption of the two goods, or save and invest - is not analysed. It is, in turn, assumed that the owners o f the specific factors would behave nonstrategically as consumers.
This stylization implies that the two players bargain in terms of rental payoffs, or in terms of relative price or in terms o f tariff rate but not in terms o f utilities as the standard Nash bargaining process is formulated. In this sense the bargaining game that is studied here is an adapted version o f the standard Nash bargaining game. Binmore (1987c) has studied such an adapted game for an exchange economy, and he preferred to call it a bartering game. In his bartering game players communicate not in terms of utilities but in terms o f real quantities o f commodities that are to be traded. We follow Binmore in this respect because it is more sensible to assume that players have perfect information on each other's rental prospects at different prices than to assume that they know each other’s rental prospects and the utilities of the rental incomes at various prices as well. Furthermore, the assumption that production sectors strive for maximization o f profit is fairly standard, and is quite common in studies o f labour market that use bargain theoretic approach (for example, Datt, 1989).
Nash (1950, 1953), who studied 2-person bargaining games, defined a solution of a bargaining game in B as a function /i:B —» R2 such that for any (9S,IT*) e B we have
ß(yi,Tld)
e SR. In other words, a solution of the game is a rule for assigning a feasible payoff to each game. Roth (1979) has argued that this rule can also be interpreted as an arbitration procedure.Nash (1953), in his seminal paper, used both strategic and axiomatic approaches to analyse the bargaining problem. In the strategic approach, he constructed a
negotiation model, so-called "demand game", in which each player (in a 2-person game) demands a particular utility payoff. If the demands of both the players are jointly compatible, then each one gets what is demanded; otherwise each receives the
predetermined disagreement payoff.4 Nash has shown that rational bargainers will agree on the payoff division that maximizes the Nash product over the feasible set.
In the axiomatic approach, Nash (1953) listed a set of eight general properties that any reasonable solution to the bargaining problem should possess. A deduction of logical solutions that satisfy the "desirable" properties led Nash to the same solution that he obtained from the negotiation model - that the solution should maximize the Nash product! Nash, therefore, wrote,
It is rather significant that this quite different approach yields the same solution. This indicates that the solution is appropriate for a wider variety of situations than those which satisfy the assumptions we made in the approach via the model (p. 136).
Subsequent authors have improved upon Nash's work and have neatly summarized his works. For example, Binmore (1987a) has listed "somewhat freely adapted versions of Nash's axioms" as follows: