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Social Choice and Welfare

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25-01-2016 | Original Paper

On surplus-sharing in partnerships

Authors: Özgür Kıbrıs, Arzu Kıbrıs

Published in: Social Choice and Welfare | Issue 1/2016

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Abstract

For investment or professional service partnerships (in general, for partnerships where measures of the partners’ contributions are available), we consider a family of partnership agreements commonly used in real life. They allocate a fixed fraction of the surplus equally and the remains, proportional to contributions; and they allow this fraction to depend on whether the surplus is positive or negative. We analyze the implications of such partnership agreements on (i) whether the partnership forms in the first place, and if it does, (ii) the partners’ contributions as well as (iii) their welfare. We then inquire which partnership agreements are productively efficient (i.e. maximizes the partners’ total contributions) and which are socially efficient, (i.e. maximizes the partners’ social welfare as formulated by the two seminal measures of egalitarianism and utilitarianism).

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Legal regulations on partnerships recognize the usage of different surplus sharing rules (as in fixing different fractions) in cases of positive and negative surplus. For example see “The practice managers’ guide to co-ownership agreements, partnerships, and associateships”, a guide for medical practices in Australia, prepared by “McMasters’ Training Pty Ltd”, available online at http://www.medicalpracticemanagement.com.au/practice_manager_s_guides/guide5/guide_5.

While we work under different informational assumptions, Holmström’s question is similar to this paper. Quoting (p. 326): “The question is whether there is a way of fully allocating the joint outcome so that the resulting noncooperative game among the agents has a Pareto optimal Nash equilibrium.” Holmström shows that the free rider problem can be solved as follows. One sets an output objective (by utilizing the observable information about the agents’ costs of effort). If it is not met, all partners receive zero as punishment. Otherwise, they share the produced value.

The parameter \(\gamma \) (respectively, \(\alpha \)) determines which fraction of positive (respectively, negative) surplus is allocated proportionally.

The full survey data is available from the University of Michigan based Inter-university Consortium for Political and Social Research (ICPSR) at their webpage: http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/8975.

Billable hours do not include administrative or clerical work or working on client development (Lang and Gordon 1995; p. 621). Therefore, the billable hours data is an understatement of a lawyer’s contribution to the partnership.

The left hand side expression \(\left( \gamma r+\left( 1-\gamma \right) \frac{ r}{n}\right) \) has two parts. The \(\gamma \) weighted part r is the partner’s return on unit contribution under proportional surplus-sharing and the \(\left( 1-\gamma \right) \) weighted part \(\frac{r}{n}\) is his return under equal surplus-sharing. The right hand side expression \(\left( \alpha \left( 1-\beta \right) +\left( 1-\alpha \right) \frac{\left( 1-\beta \right) }{n}\right) \) again has two parts. The \(\alpha \) weighted part of this expression, \(\left( 1-\beta \right) \) is the loss incurred for unit contribution in case of proportional surplus-sharing and the \(\left( 1-\alpha \right) \) weighted part \(\frac{1-\beta }{n}\) is the loss incurred in case of equal surplus-sharing.

It follows from Eq. (3) in the “Appendix” that the partnership games induced by \(PE\left[ \gamma ,\alpha \right] \) agreements admit dominant strategy equilibria if and only if \(\left( 1-\gamma \right) r+\left( 1-\alpha \right) \left( 1-\beta \right) =0\) (in which case, partner i’s best response is independent of the others’ strategies). This equality holds if and only if \(\alpha =\gamma =1\). In Eq. (1), this equality ensures that partner i ’s equilibrium strategy is independent of the others’ risk attitudes.

As can be more formally seen in Eq. (3) in the “Appendix”, both partners have linear best response functions (with a positive intercept and a negative slope). An increase in \(\alpha \) affects both best response functions in the same way: it decreases the intercept and decreases the slope in absolute value, making best responses less sensitive to the other partner’s choices. It is because of this that the strategic substitutes property matters less at high values of \(\alpha .\)

Figure 2 also demonstrates that, for \(\gamma \le 0.1,\) the partnership agreement \(PE\left[ \gamma ,\alpha \right] \) is not acceptable and, as discussed in the previous section, the partnership does not form.

To see this, note that \(\frac{\frac{1}{a_{n}}}{\sum _{N}\frac{1}{a_{j}}}>1- \frac{r+\alpha \left( 1-\beta \right) }{r+1-\beta }\) iff \(\alpha >\frac{ \left( r+1-\beta \right) }{\left( 1-\beta \right) }\left( 1-\frac{\frac{1}{ a_{n}}}{\sum _{N}\frac{1}{a_{j}}}\right) -\frac{r}{\left( 1-\beta \right) }\) iff \(\alpha >1-\frac{r+1-\beta }{1-\beta }\frac{\frac{1}{a_{n}}}{\sum _{N} \frac{1}{a_{j}}}\).

This corresponds to 1,324,692 combinations of \(\alpha ,\gamma , \beta ,p,r,a_{1},a_{2}\).

This corresponds to 1,029,914 combinations of \(\alpha ,\gamma , \beta ,p,r,a_{1},a_{2},a_{3}\).

There is a second indirect effect on individual contributions stemming from the fact that the partners’ contributions are strategic substitutes. A partner increasing his contribution incentivizes the other partners to decrease their contributions in return. A combination of these two effects can thus create nonmonotonic individual contribution responses to changes in \(\alpha \) and \(\gamma \) as seen in Figs. 1 and 2. Yet, as our theorem shows, when aggregated over agents, this first effect overrides the second.

This expression is equal to 0 if and only if \(\alpha =\gamma =1\) and equal to 1 if and only if \(\alpha =\gamma =0.\) The former is trivial. To see the latter, note that \(\frac{r\left( 1-\gamma \right) +\left( 1-\beta \right) \left( 1-\alpha \right) }{\left( 1-\alpha +n\alpha \right) \left( 1-\beta \right) +\left( n-1\right) r\gamma +r}\le 1\) simplifies to \(0\le n\alpha \left( 1-\beta \right) +nr\gamma ,\) achieved with equality if and only if \( \alpha =\gamma =0\).

Note that, this expression gives total contribution when \(\alpha =\gamma =0\) as well. Even though there is multiplicity of equilibria in this case, they all have the same total contribution level given by this expression.

For brevity of presentation, calculations that prove this and similar secondary claims have been skipped. However, they all are available from the authors upon request.

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Title: On surplus-sharing in partnerships
Authors: Özgür Kıbrıs
Arzu Kıbrıs
Publication date: 25-01-2016
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 1/2016
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0947-7

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