Divide and Invest: Bargaining in a Dynamic Framework
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Divide and Invest: Bargaining in a Dynamic Framework Francesca Flamini1 Received: 29 November 2019 / Accepted: 23 October 2020 / Published online: 23 November 2020 © The Author(s) 2020
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. Keywords Bargaining · Investment · Recursive optimization · Markov perfect equilibrium I wish to thank the editors, two anonymous referees and the participants of the conferences of the European Economic Association (Glasgow), the Royal Economic Society (Warwick) and Game Theory Society (Maastricht) for useful comments and suggestions. Financial support by the Economic and Social Research Council (Grant no. RES-061-23-0084) is gratefully acknowledged. All errors remain mine. * Francesca Flamini [email protected] 1
Economics, Adam Smith Business School, University of Glasgow, Glasgow G12 8QQ, UK
13
Vol.:(0123456789)
122
Homo Oeconomicus (2020) 37:121–153
JEL Classification C73 · C78 · D9
1 Introduction Several bargaining situations, in the most diverse contexts, are dynamic, in that current agreement may affect future bargaining possibilities. For instance, partners in a business need to negotiate not only over how to split profits among themselves, but also over how much profit should be re-invested for the following production period, knowing that their level of investment today affects the size of production/profit tomorrow. Countries may attempt to
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