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Review of Accounting Studies

Erschienen in:

13.01.2020

Incentives in optimally sized teams for projects with uncertain returns

verfasst von: Oliver Dürr, Markus Nisch, Anna Rohlfing-Bastian

Erschienen in: Review of Accounting Studies | Ausgabe 1/2020

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Abstract

This paper analyzes a principal-agent model with three risk-averse players to investigate incentive provision and optimal team size in a setting with uncertain productivity and team synergies. A principal hires a team of workers and a manager to supervise the team. Workers provide productive effort, whereas the manager exerts effort to reduce measurement noise and productivity risk. We find that moral hazard is a limiting factor for team size and that the risk from uncertain productivity leads to smaller optimal teams, which stands in contrast to previous literature. Furthermore, we show that the manager’s and workers’ compensation increases with team size and that the pay differential between them is higher for larger teams. Our analysis demonstrates that the interdependency between team size and incentive provision makes it essential to coordinate the choice of these design variables.

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Chandrasekaran and Linderman (2015) observe a negative correlation between team size and project complexity, which can be interpreted as a proxy for the uncertain productivity of a project. Dailey (1978) and van de Ven et al. (1976) show a negative correlation between task uncertainty and team size.

For an overview with a particular focus on accounting applications, see Lambert (2001).

Liang et al. (2008) cannot derive a first-best solution. Due to their model construction, the optimal team size in a first-best setting would tend toward infinity.

Vectors are in boldface, and T denotes the transpose of a column vector.

Liang et al. (2008) capture the trade-off for larger teams as a positive linear effect on output and an increase in performance measure risk as a negative component.

Of course, similar examples can be found in other industries. In the IT industry, the success of software development is uncertain. In banking, the ongoing process of digital transformation has an uncertain outcome.

For simplicity, we abstract from the assumption of Liang et al. (2008) that measurement noise increases with team size. However, incorporating such an assumption would not change the results qualitatively.

The variance σ_x can be approximated by a binomial distribution for large N, that is, σ_x = N ⋅ q ⋅ (1 − q), where N is the number of customers and q is the probability with which customers will buy the firm’s product. Reducing σ_x through a market research effort s would imply increasing the probability q of each potential customer buying the product and reducing the number of potential customers N as preferences are more clearly identified and some customer groups drop out.

Henceforth, we omit the index i in the parameters for the workers.

Installation of a monitoring system, analysis of market prospects, and development of a project are independent tasks such that none of them requires another task to be finished first. We therefore assume that the workers’ and manager’s efforts are exerted simultaneously.

To reduce the complexity of the model, we disregard the time value of money.

We present a variation of the model with a risk-neutral principal in Section 5.

The results presented in the following also hold for different risk-aversion coefficients for the manager and the workers. We have used the simplified assumption of identical risk-aversion to make our model comparable to Liang et al. (2008).

Liang et al. (2008) assume a risk-neutral principal. Consequently, they do not establish a first-best solution to their model because team size would be optimally set to infinity if efforts were observable.

In the model of Liang et al. (2008), the manager only exerts effort m, and the manager’s effort cost is C(m) = k ⋅ m. We assume k = 1 and add the second effort (market research). The tasks are independent, so we neglect potential spillovers in the effort cost functions.

See Appendix A for a presentation of the principal’s optimization program.

All proofs can be found in Appendix A.

Setting λ = 0 results in the standard solution of a risk-neutral principal with \(\beta _{x}^{*}=0\), where the entire accounting income risk is borne by the principal and the manager is paid solely a fixed salary. See also Section 5.

In the first-best solution (see the proof of Lemma 1 in Appendix A), the workers’ and manager’s effort choices are interdependent, because they are connected in the principal’s certainty equivalent. In the second-best scenario, this is no longer the case, since the effort choices are determined based on the individual certainty equivalents of the workers and the manager. The manager anticipates the workers’ optimal effort choice, since their productive efforts determine the level of risk exposure from the uncertain productivity. The market research effort s is then chosen strategically, considering the workers’ optimal productive effort choice.

Note that indeed \(\beta _{y}^{\dagger }=\frac {\sqrt {n}\sqrt {r+n \lambda }}{\sqrt {2 r+\lambda }}\cdot b^{\dagger }\).

We will elaborate on this issue in Proposition 4.

This result comports with the result in Proposition 4 of Liang et al. (2008).

Another difference in their model is the assumption of a risk-neutral principal. However, the positive impact of risk on team size also holds for a risk-averse principal. See Appendix A for a proof of this statement.

The results are robust to different parameter values and relations among them as well as for different functional forms for the synergy effect and the organizational costs.

See Proposition 5 in Liang et al. (2008).

Note that the manager’s total compensation in the second-best setting needs to be paid for also performing effort m, which was not necessary in the first-best setting. Therefore the change in the manager’s total compensation is only very small from first to second best.

The results are also robust to variations in the relations of the underlying parameter values.

For the ease of notation, we omit the optimality index ‡ in the following derivation of the proof.

In the work of Liang et al. (2008), \(\beta ^{*}=\sqrt {\frac {n}{2}}\cdot b^{*}\), which corresponds to our result if we assume a risk-neutral principal, that is, λ = 0.

The notation is the same as for Fu et al. (2016), with λ as the principal’s risk aversion.

The numbering of equations refers to Fu et al. (2016)

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Titel: Incentives in optimally sized teams for projects with uncertain returns
verfasst von: Oliver Dürr
Markus Nisch
Anna Rohlfing-Bastian
Publikationsdatum: 13.01.2020
Verlag: Springer US
Erschienen in: Review of Accounting Studies / Ausgabe 1/2020
Print ISSN: 1380-6653
Elektronische ISSN: 1573-7136
DOI: https://doi.org/10.1007/s11142-019-09529-5

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