Bargaining under liquidity constraints: Unified strategic foundations of the Nash and Kalai solutions

We provide unified strategic foundations for the Nash (1950) and Kalai (1977) solutions in the context of negotiations under liquidity constraints. We propose an N-round game where in each round a seller and a buyer with limited payment capacity negotiate a bundle of divisible goods, where bundle sizes can vary across rounds, according to Rubinstein's (1982) alternating-offer game. The game implements the Nash solution if N=1 and the Kalai solution if N=+∞ and bundle sizes are infinitesimal. If N is set by one player ex ante, the buyer chooses N=1 while the seller chooses N=+∞. We endogenize liquidity constraints and show they bind for all N<+∞, even when there is no cost in holding liquidity.