Controlling Scarce Working Time in a Multi-task Incentive Problem^§

§ This paper was previously titled: Congruent allocation of scarce managerial time in a multi-task contracting problem.

Abstract

Even though working time is one of the firm's scarcest resources, there are very few insights into how to optimally control it. In this paper we analyze the optimal allocation of scarce working time to two tasks in a contracting problem. Since each hour devoted to one specific task is not available for the other task, and vice versa, the agent accounts for opportunity cost when allocating his time to the tasks. We analyze this trade-off assuming that the agent's contribution is not contractible, such that a contract must be written based on an alternative measure. Time scarcity restricts total output. At the same time, however, it changes the agent's focus of attention towards the task with the higher performance sensitivity. Thus, by increasing the incentive coefficient the principal induces the agent to shift time from the less sensitive to the more sensitive task. The main contribution of our paper is that we show that time scarcity can improve the congruence of the agent's behavior with the principal's objectives. Under certain conditions an improved alignment of effort under time scarcity offsets the reduction of output and increases the agency's surplus.

Acknowledgements

We would like to thank Christian Hofmann (the editor), two anonymous reviewers, Sandra K. Kronenberger, Barbara Schöndube-Pirchegger and the participants of the 37th Annual Congress of the European Accounting Association in Tallinn for very helpful comments.

Notes

§ This paper was previously titled: Congruent allocation of scarce managerial time in a multi-task contracting problem.

1 For example, Bevins & De Smet (2013) emphasize that for executives the ‘perennial time-scarcity problem […] has become more acute in recent years’ (p. 28). Furthermore, according to the report of The Economist Intelligence Unit (2015), 45% of executives view time constraints as the biggest problem in achieving professional goals.

2 Although the agent is risk neutral, there is a non-trivial agency conflict in our analysis as the agent's contribution to firm value is non-contractible. See Baker (1992) and Baker, Gibbons, & Murphy (1994) for similar approaches under non-contractability.

3 For example, the empirical evidence that the executives' pay-performance sensitivity is unexpectedly low (see Jensen & Murphy, 1990).

4 Usually, full-time means 35 or more hours per week and part-time less than 35 hours per week, see, for example, the Bureau of Labor Statistics at https://www.bls.gov/cps/lfcharacteristics.htm#fullpart.

5 Multi-tasking approaches under scarcity are also studied in the tournament literature, see Kvasov (2007), Strömberg (2008), and Mauch (2014).

6 Related issues are analyzed by Banker & Thevaranjan (1997), Budde (2007) and Liang & Nan (2014). Although Baker (1992) and Bushman, Indjejikian, & Penno (2000) consider single action models, their settings can also be regarded as multi-task problems since the principal has to allocate the agent's action across different states of the world.

7 Overload here means that the agent is induced to exert inefficiently high effort for an additional project in order to reduce his rent from the other projects.

8 We refer to π also simply as the ‘firm value’ or the ‘output’ in what follows.

9 Our results do also hold true if one task would be regarded as a sabotage activity, that is, if one of the parameters $g_X$ or $g_Y$ would be negative. As including this into the analysis results in several additional corner solutions to be distinguished, we restrict both parameters to be non-negative.

10 Similar to Budde (2007) we assume that the agent's contribution to firm value π cannot be separated from the firm's other assets and therefore cannot be rented (sold) in isolation.

11 The agent could ignore his sleep and regeneration needs in the short run, but not permanently. Over the year he needs a minimum amount of rest time.

12 Instead of using a cost function $C(e_1+e_2)$ we assume $C\left(e_1\right)+C\left(e_2\right)$ as, for example, Budde (2007) and Feltham & Xie (1994). This assumption is reasonable if task fulfillment gets more complicated as completion comes closer. People tend to work on the easy parts of the tasks first to achieve first results and leave the more complex parts for later points in time. Spending one additional hour on a task to which has been devoted a lot of time is then more costly than spending it on a task which is still in the early stages. In the extension in Section 6.1 we consider partially substitutable efforts in the agent's cost function.

13 We denote optimal solutions under the time constraint with $(\cdot ,t)$.

14 $v=\mathrm{\Delta }g/\mathrm{\Delta }b$ is not always the unique optimal incentive coefficient, given $\mathrm{\Delta }g/\mathrm{\Delta }b\ge \tau$. If $v=\mathrm{\Delta }g/\mathrm{\Delta }b$ induces a corner solution (which happens if $\mathrm{\Delta }g/\mathrm{\Delta }b\ge T/\left|\mathrm{\Delta }b\right|)$ all incentive rates $v\ge T/\left|\mathrm{\Delta }b\right|$ induce the same corner solution, too.

15 $v^{\ast }$ can also be written as $v^{\ast }=(\sqrt{g_X^2+g_Y^2}/\sqrt{b_X^2+b_Y^2})\cos \left(\alpha \right)$ with α as the angle between the vectors $(b_X,b_Y)$ and $(g_X,g_Y)$. $\cos \left(\alpha \right)$ is a measure of distortion of the performance measure and the first term scales the length of the two vectors, see Baker (2002) for details. High distortion (low $\cos \left(\alpha \right))$ leads to low powered incentives.

16 In a pure effort allocation problem, the principal minimizes the distance between the first-best and the implementable second-best effort, see, for example, Datar et al. (2001, p. 80). To see this consider Figure 1(a) where the point $(e_X^{\mathrm{FB}},e_Y^{\mathrm{FB}})$ represents the first-best effort and the point $(e_X^{\ast },e_Y^{\ast })$ has the shortest distance to $(e_X^{\mathrm{FB}},e_Y^{\mathrm{FB}})$. In the figure the set of implementable second-best efforts is given by the line denoted by $b_Y/b_X$. Thus, when comparing the second-best effort allocation with and without time constraint, instead of the surpluses we can compare the distances from the (unconstrained) first-best solution.

17 See the proof of Proposition 4 for details.

18 In the terminology of Feltham & Wu (2000), in this case the performance measures are not perfectly aligned with each other.

19 Notice that if a signal's relative sensitivity in terms of task j is too high, for example, $a_j/a_{-j}>g_j/g_{-j}$, the relative sensitivity in terms of task $-j$ is too low, $a_{-j}/a_j<g_{-j}/g_j$.

20 More generally, there are restrictions by law of using the data from time tracking of workers. For example, in Germany limitations follow from the Works Council Constitution Act (Betriebsverfassungsgesetz).

21 While this is not possible for e.g. agricultural workers, many white-collar workers are able to work on their tasks at home.

22 To be precise, we assume $0<\delta <min\{b_X/b_Y,b_Y/b_X\}(<1)$ to ensure that no negative effort is induced in one task to reduce the agent's disutility of working.

23 For example, consider the question whether estimated pay-performance sensitivity is too low or appropriate (see Hall & Liebmann, 1998; Jensen & Murphy, 1990) and the related discussion of the interplay between the pay-performance sensitivity and the marginal product of effort (Baker & Hall, 2004). Or, more generally, the mixed evidence about the factors that substantially affect the provision of incentives. Besides noise and risk aversion as, for example, critically discussed by Prendergast (1999), our study suggests that time scarcity may be regarded as another factor constraining incentive provision of workers and managers.

24 Applying Proposition 1 to $e_X+e_Y\ge T$ yields $e_X^{\mathrm{FB},t}=(T+\mathrm{\Delta }g)/2$ and $e_Y^{\mathrm{FB},t}=T-e_X$. As $\mathrm{\Delta }g<T$ corner solutions cannot occur.