# Bargaining when T is infinite

What about the situation in which there is no artificially imposed point in time for the bargaining to end? In other words, what happens as *T **^* ~? One reasonable 'guess' is that the payoffs in the infinite horizon bargaining situation are the limits as *T **^* ~ of the payoffs in the

1+ _{S}^{T +1}

simpler case in which *T* < <*>. Since *S **<* 1, the limit of *n* as *T* ^ »

1 + *S*

is . The residents' payoff is whatever is left over, which is:

1*+**S
*

Conveniently, this 'guess' turns out to be correct: the unique equilibrium payoffs of the infinite horizon version of this game are the limit of the equilibrium offers in the finite horizon game. Moreover, the unique equilibrium involves the residents immediately accepting the factory's offer. Again, since haggling is allowed and the parties are impatient, the factory's offer is immediately accepted.

# Non-cooperative bargaining without frictions: The split the surplus rule

Recall that in the non-cooperative bargaining framework, if a player receives a share of the surplus *x** _{i}* in period

*t*, then the benefit from that share is:

where *d*_{t} = *e**~ ^{r}‘^{A}* is party i's discount factor and A > 0 is the absolute size of the frictions in the bargaining process. As frictions disappear, A ^ 0, which implies that:

Applying the result in equation (11.7), this then implies that as bargaining becomes frictionless, the equilibrium shares of the parties con-

*л*

verge to *n*_{F} = *n** _{R} =* . That is, if the parties have identical discount rates

and if bargaining frictions are non-existent, the factory's first mover advantage disappears and the parties will 'split the surplus' equally. We will return to this result.