Divide and Invest: Bargaining in a Dynamic Framework

Abstract

Many negotiations (for instance, among political parties or partners in a business) are characterized by dynamic bargaining: current agreements affect future bargaining possibilities. We study such situations using bargaining games á la Rubinstein (Econometrica 50:97–109, 1982), with the novelty that players can decide how much to invest, as well as how to share the residual surplus for their own consumption. Their investment decisions affect the size of the next surplus. In line with the existing literature, we focus on Markov Perfect Equilibria, where consumption and investment are linear time-invariant functions of capital and show that standard results in bargaining theory can be overturned. For instance, a more patient proposer may consume less than his opponent. The intuition is that when capital is productive, both parties have incentives to invest, however, the most patient party wishes to invest significantly more than his opponent. Then, to prioritize investment—which affects future bargaining possibilities—the former must make larger concessions and let the latter consume more. Another interesting result is that if a player becomes more patient, both parties may reduce their investment. The key underlying driver of this result is that when counteroffers become cheaper for an impatient party, he is able to reduce his investment and consume more. This forces his opponent to make larger concessions (and reduce his investment plan). Moreover, extreme demands (where a player consumes all the residual surplus) are possible in equilibrium, under fairly modest assumptions. Finally, only when bargaining is frictionless, is the equilibrium efficient.

The Role of the Status-Quo in Dynamic Bargaining

“…For T = ∞, the state variables are the status-quo, s t (which is either the latest agreement, if any, or the initial status-quo, s), and the capital stock, k t . If players agree to change the status-quo, then it must be that the proposer gains, see (4), and using (7), the responder is also (weakly) better off. Hence, the status-quo is implemented, if Pareto efficient, otherwise players can move to a Pareto superior outcome.…”

“…First, while in [1] the surplus is fixed in every period, in our framework, current policy choices (in particular, on how much to invest) will determine future surpluses 6 . This is relevant, as often major economic reforms (e.g., a more flexible labour market 7 ) can be initially costly but could have longterm benefits (as it stimulates the economy and leads to higher future budgets). Second, in our model players may differ in terms of time preferences, while in [1], players are equally patient (but put different weights on the public good).…”

“…The closest papers in dynamic bargaining are [4,7] (Section 10). The latter is the first contribution with a focus on a repeated (non-cooperative) bargaining game with investment decisions in addition to the standard consumption decisions.…”

“…Ref. [7] has the main aim of addressing the problem of how much parties invest in a strategic framework. In this paper, we adopt this framework with two (potentially heterogeneous) players who need to choose both how much to consume and how much to invest, but, in addition, we endogenize the status-quo.…”