Incentives in optimally sized teams for projects with uncertain returns

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.

Appendices

Appendix A

The principal’s optimization problem is

$$ \begin{array}{@{}rcl@{}} \max_{b,\beta_{x},\beta_{y},n} \phantom{...} & & CEP(\cdot)\\ \text{s.t.} \phantom{...} & & CEW(\cdot)\geq0, \qquad\qquad\quad \ \ \qquad\qquad\qquad(IRW)\\ && CEM(\cdot)\geq0, \qquad \qquad\quad\ \ \qquad\qquad\qquad(IRM)\\ && e\in\operatornamewithlimits{argmax}_{e'\in\mathbb{R}}\phantom{..}CEW(\cdot), \qquad \qquad\qquad\qquad(ICW)\\ && s\in\operatornamewithlimits{argmax}_{s'\in\mathbb{R}}\phantom{..}CEM(\cdot), \qquad \qquad\qquad\qquad(ICM_{1})\\ && m\in\operatornamewithlimits{argmax}_{m'\in\mathbb{R}}\phantom{..}CEM(\cdot). \qquad \qquad\qquad\qquad(ICM_{2}) \end{array} $$

Proof Proof of Lemma 1.

If efforts are observable, the optimal effort choices for the workers and the manager are determined by differentiating the principal’s certainty equivalent, given in Eq. 3 with respect to e, s, and m. Solving the respective first-order conditions yields

$$ \begin{array}{@{}rcl@{}} e&=&\frac{p s G(n)}{s+n G(n)^{2}({\beta_{x}^{2}} r+(1-\beta_{x})^{2}\lambda)\cdot {\sigma_{p}^{2}}}, \\ s&=& n e G(n)\sqrt{{\beta_{x}^{2}} r+(1-\beta_{x})^{2}\lambda}\cdot\sigma_{p}\text{, and} \\ m^{*}&=&\sqrt{\lambda(\beta_{y}+n b)^{2}+r({\beta_{y}^{2}}+n {b}^{2})}\cdot\sigma_{y}. \end{array} $$

Solving the system of equations given by e and s yields

$$ \begin{array}{@{}rcl@{}} e^{*}&=&G(n)\cdot\left( p-\sqrt{{\beta_{x}^{2}} r+(1-\beta_{x})^{2}{\lambda}}\cdot \sigma_{p}\right)\text{ and }\\ s^{*}&=&n G(n)^{2}\cdot\sqrt{{\beta_{x}^{2}} r+(1-\beta_{x})^{2}{\lambda}}\cdot \sigma_{p} \left( p-\sqrt{{\beta_{x}^{2}} r+(1-\beta_{x})^{2}{\lambda}}\cdot \sigma_{p}\right). \end{array} $$

Inserting the optimal effort choices as well as the binding individual rationality constraints (IRW) and (IRM) into the principal’s certainty equivalent in Eq. 3 yields

$$ \begin{array}{@{}rcl@{}} CEP &=& \frac{1}{2}n G(n)^{2}\left( p-\sqrt{\lambda-2\lambda \beta_{x}+(r+\lambda){\beta_{x}^{2}}}\cdot \sigma_{p} \right)^{2} \\ &&-\sqrt{\lambda(\beta_{y} + n b)^{2} + r ({\beta_{y}^{2}}+n b^{2})} \cdot \sigma_{y} -C(n). \end{array} $$

Differentiating this equation with respect to b, β_x, and β_y and solving the respective first-order conditions yields

$$ \begin{array}{@{}rcl@{}} b&=&-\frac{\lambda \beta_{y}}{r+n\lambda},\\ \beta_{x}^{*}&=&\frac{\lambda}{r+\lambda}, \text{ and}\\ \beta_{y}&=&-\frac{n \lambda b}{r+\lambda}. \end{array} $$

Solving the systems of equations given by b and β_y leads to optimal incentive rates of $b^{*}=\beta _{y}^{*}=0$ and the respective incentive rate $\beta _{x}^{*}$ given in the lemma. Inserting the optimal incentive rates into the principal’s payoff yields

$$ {CEP}^{*}= \frac{1}{2} n G(n)^{2} \left( p-\sqrt{\frac{r \cdot \lambda}{r + \lambda}}\cdot \sigma_{p}\right)^{2} - C(n). $$

□

Proof Proof of Lemma 2.

Differentiating Eq. 1 with respect to e and Eq. 2 with respect to s and m yields the optimal effort choices

$$ \begin{array}{@{}rcl@{}} e^{\dagger}&=&G(n)b,\\ s^{\dagger}&=&n G(n)^{2}\sqrt{r} \beta_{x} b \sigma_{p}, \text{ and}\\ m^{\dagger}&=&\sqrt{r} \beta_{y} \sigma_{y}. \end{array} $$

Inserting the optimal effort choices into the principal’s certainty equivalent Eq. 3 and considering the binding individual rationality constraints (IRW) and (IRM) yields

$$ \begin{array}{@{}rcl@{}} CEP &=& n b G(n)^{2} \bigg(p - \frac{1}{2\sqrt{r}\beta_{x}}(\lambda (1 - \beta_{x})^{2} + 2 r {\beta_{x}^{2}})\cdot \sigma_{p} \bigg) -\frac{1}{2}n G(n)^{2} b^{2} \\ &-& \frac{1}{2\sqrt{r}\beta_{y}}\cdot \sigma_{y} \bigg(n (n \lambda + r) b^{2} + 2 n \lambda b \beta_{y} + (2 r + \lambda) {\beta_{y}^{2}}\bigg) -C(n). \end{array} $$

Differentiating this expression with respect to b, β_x, and β_y and solving the respective first-order conditions gives the optimal incentive rates

$$ \begin{array}{@{}rcl@{}} b&=&\frac{\beta_{y}}{(r+n\lambda)\sigma_{y} + \sqrt{r}\beta_{y} G(n)^{2}}\!\left( \!G(n)^{2}\!\left( p\sqrt{r} - (r \beta_{x}+\frac{\lambda(1-\beta_{x})^{2}}{2\beta_{x}})\sigma_{p}\right)-\lambda\sigma_{y}\right), \\ \beta_{x}^{\dagger}&=&\frac{\sqrt{\lambda}}{\sqrt{2r+\lambda}}\text{, and} \\ \beta_{y}&=&\frac{\sqrt{n}\sqrt{n \lambda+r}}{\sqrt{2r+\lambda}}b. \end{array} $$

Solving the respective system of equations given by b and β_y yields the optimal second-best incentive rates stated in the lemma (expressions Eqs. 5, 6 and 7). Inserting the optimal incentive rates into the principal’s utility gives

$$ {CEP}^{\dagger}=\frac{1}{2} n G(n)^{2}\left[p-\psi^{\dagger}-\theta^{\dagger}(n)\right]^{2}- C(n). $$

□

Proof Proof of Proposition 1.

We first obtain the conditions for the optimal team size in the first- and second-best settings. Differentiating the principal’s utilities derived in connection with Lemmas 1 and 2 with respect to n yields the following necessary conditions for the optimal team size:

$$ \begin{array}{@{}rcl@{}} (p-\psi^{*})^{2} G(n^{*})\bigg(\frac{1}{2}G(n^{*}) +n^{*} G'(n^{*}) \bigg) - C'(n^{*})&\!=&\!0,\\ G(n^{\dagger}) \left( \frac{1}{2}G(n^{\dagger})+n^{\dagger} G'(n^{\dagger})\right)(p - \psi^{\dagger}\!-\theta^{\dagger}(n^{\dagger}))^{2}&&\\ + \!\left( \!\frac{1}{2}\sigma_{y}\frac{\sqrt{r}}{\sqrt{n^{\dagger}}}\frac{\sqrt{2r + \lambda}}{\sqrt{r + n^{\dagger} \lambda}}+2n^{\dagger} G'(n^{\dagger})G(n^{\dagger})\theta^{\dagger}(n^{\dagger})\!\right)(p - \psi^{\dagger} - \theta^{\dagger}(n^{\dagger})) - C'(n^{\dagger})&\!=&\!0, \end{array} $$

with $\psi ^{*}=\sqrt {\frac {r\cdot \lambda }{r + \lambda }}\cdot \sigma _{p}$ and ψ^‡ as well as 𝜃^‡ as defined in Lemma 2. The sufficient condition is obtained by formulating the second derivative, which is exemplarily done for the first-best case but works the same for the second-best case:

$$ \frac{\partial^{2} {CEP}^{*}}{\partial n^{2}} = \underbrace{\left( p - {\psi^{*}}\right)^{2}}_{\text{I}}\cdot \underbrace{\!\bigg(\!G(n^{*}) G'(n^{*}) + n^{*} G'(n^{*})^{2} + n^{*} G(n^{*}) G^{\prime\prime}(n^{*}) \!\bigg)}_{\text{II}} - \underbrace{C^{\prime\prime}(n^{*})}_{\text{III}}\!<\!0. $$

Part I of the term is positive, and part II can either be positive or negative, depending on the functional form of G(n). Part III is negative, as assumed from the functional form of the organizational costs. For an extremely large team size ($n \to \infty$), the product I ⋅ II will approach zero, but the organizational costs will approach $-\infty$, so that the existence of a maximizing optimum is ensured as long as the risk-adjusted productivity and the marginal synergy effect are sufficiently large for n = 1. We can now compare the two conditions for the optimal team size in the first- and second-best case. This allows us to make an inference on the relation between the first- and second-best team sizes:

$$ \begin{array}{@{}rcl@{}} \frac{\partial {CEP}^{*}}{\partial n}&=& -C'(n) + \underbrace{(p-\psi^{*})^{2} G(n)\bigg(n G'(n) + \frac{1}{2}G(n) \bigg)}_{\text{I}}, \\ \frac{\partial {CEP}^{\dagger}}{\partial n}&=& -C'(n) + \underbrace{(p - {\psi^{\dagger}})^{2} G(n) \bigg(n G'(n) + \frac{1}{2} G(n) \bigg)}_{\text{I'}} \\ && - \underbrace{{\theta^{\dagger}(n)}^{2} G(n) \bigg(n G'(n) + \frac{1}{2}G(n) \bigg) }_{\text{II}}\\ & &- \underbrace{(p-{\psi^{\dagger}}-{\theta^{\dagger}(n)}) {G(n)^{2}} {\theta^{\dagger}(n)} \left( \frac{2r + \lambda + 4 n \lambda + \frac{\sqrt{n} \lambda \sqrt{2r + \lambda} }{\sqrt{r + n \lambda}} }{4 r + 2 \lambda +4 n \lambda} \right)}_{\text{III}}. \end{array} $$

The optimal team size is determined by the intersection of the marginal cost function with the marginal benefit function. The marginal cost function C^′(n) is increasing at an increasing rate and identical in the first- and second-best cases. The marginal benefits are increasing at a decreasing rate. The marginal benefits in the first-best case are given by part I and by parts I’, II, and III in the second-best case. Apart from the parameter ψ, parts I and I’ are identical. We next need to show that ψ^‡ > ψ^∗. We prove the inequality by contradiction, that is, we start with ψ^‡ < ψ^∗ and show that this leads to a conflict with a basic assumption on the underlying model parameters. Replacing the simplification symbols ψ^∗ and ψ^‡ with the respective terms yields the initial inequality:

$$\frac{\sqrt{\lambda}}{\sqrt{r}} \bigg(\sqrt{2 r + \lambda} - \sqrt{\lambda}\bigg) < \frac{\sqrt{r\cdot \lambda}}{\sqrt{r + \lambda}}.$$

We then simplify and rearrange the terms to eliminate the fractions:

$$\sqrt{r + \lambda} \sqrt{2 r + \lambda} - \sqrt{\lambda} \sqrt{r + \lambda} < r.$$

We then square the inequality and simplify the terms:

$$ \begin{array}{@{}rcl@{}} 2 r^{2} + 3 r \lambda + \lambda^{2} & <& r^{2} + 2 r \sqrt{\lambda r + \lambda^{2}} + \lambda r + \lambda^{2}, \\ r + 2 \lambda & < &2 \sqrt{\lambda r + \lambda^{2}}, \\ r^{2} + 4 r \lambda + 4 \lambda^{2} & <& 4 \lambda r + 4 \lambda^{2},\\ r^{2} & < &0. \end{array} $$

Risk aversion is assumed to be positive, and the squared value needs to be positive in any way. We therefore obtain a contradiction and have proven that ψ^‡ > ψ^∗. With this result, it follows that I > I’; that is, this part of the marginal benefits is smaller in the second- than in the first-best case. Moreover, parts II and III are positive and enter the equation in the second-best case with a negative sign, such that the marginal benefits are further reduced in the second-best case. We can thus conclude that the marginal benefit function in the second-best case is below the marginal benefit function in the first-best case. Given the functional specifications, the intersection of the second-best marginal benefit function with the marginal cost function, $\displaystyle \frac {\partial {CEP}^{\dagger }}{\partial n}=0\mid _{n=n^{\dagger }}$, needs to occur for smaller n than the intersection of the first-best marginal benefit function with the marginal cost function, $\displaystyle \frac {\partial {CEP}^{*}}{\partial n}=0\mid _{n=n^{*}}$; that is, n^‡ < n^∗. □

Proof Proof of Proposition 2.

The comparative statics of the optimal team size n^‡ with respect to any exogenous parameter are derived by implicit differentiation. We denote the exogenous parameters p, σ_p, and σ_y by h. From the optimality condition in the proof of Proposition 1, we define

$$ \frac{\partial {CEP}^{\dagger}}{\partial n}=F(n,h)\stackrel{!}{=} 0. $$

From the rule for implicit differentiation, we obtain

$$ \frac{\partial F}{\partial h} + \frac{\partial F}{\partial n^{\dagger}} \cdot \frac{\partial n^{\dagger}}{\partial h} = 0. $$

Rearranging the terms leads to

$$ \frac{\partial n^{\dagger}}{\partial h} = -\frac{\frac{\partial F}{\partial h}}{\frac{\partial F}{\partial n^{\dagger}}}=-\frac{F_{h}}{F_{n^{\dagger}}}. $$

We now need to define the partial derivative of the optimality condition F with respect to the optimal team size, which is equal to the second derivative of the principal’s certainty equivalent with respect to n:

$$ \frac{\partial F}{\partial n}= \frac{\partial^{2} {CEP}^{\dagger}}{\partial n^{2}}<0. $$

This expression has already been defined as being negative (sufficient condition for a concave structure of the principal’s expected utility such that the optimal team size maximizes the principal’s utility). The direction of the effect of a change in any of the exogenous parameters, h, is therefore the same as the direction of the effect of a change in any of the exogenous parameters on the function F:

$$ sgn\left( \frac{\partial n^{\dagger}}{\partial h}\right) = -\frac{\overbrace{F_{h}}^{\text{\large{?}}}}{\underbrace{F_{n^{\dagger}}}_{\text{\LARGE{-}}}}=sgn(F_{h})=sgn\left( \frac{\partial F}{\partial h}\right). $$

The sign is determined by formulating the partial derivatives and using the assumptions for the relation of the risk aversions and the function G(n):

$$ \begin{array}{@{}rcl@{}} sgn\!\left( \!\frac{\partial n^{\dagger}}{\partial \sigma_{p}}\!\right)\! &=&\! sgn\!\left( \frac{\partial F}{\partial \sigma_{p}}\right)\\ &=& \! sgn\!\left( \! - \frac{\sqrt{\lambda}}{\sqrt{r}}(\sqrt{2r \! +\! \lambda}\! -\! \sqrt{\lambda})\bigg(\! (G(n)^{2}\! +\! 2nG'(n) G(n)) (p\! -\! \psi^{\dagger}\! -\! \theta^{\dagger}(n)) \right. \\ && \left. + \frac{1}{2} \sigma_{y} \frac{\sqrt{r}}{\sqrt{n}} \frac{\sqrt{2r+\lambda}}{\sqrt{r+n \lambda}} +2n G'(n)G(n) \theta^{\dagger}(n) \bigg) \right) \\ &=&\! -1, \end{array} $$

$$ \begin{array}{@{}rcl@{}} sgn\!\left( \!\frac{\partial n^{\dagger}}{\partial \sigma_{y}}\!\right)\! &=&\! sgn\! \left( \frac{\partial F}{\partial \sigma_{y}}\right)\\ &=&\! sgn\! \left( \! -\frac{\theta^{\dagger}\!({\kern-.5pt}n{\kern-.5pt})}{\sigma_{y}}\! \bigg(\! 2 n G'\!({\kern-.5pt}n{\kern-.5pt}){\kern-.5pt} G{\kern-.5pt}({\kern-.5pt}n{\kern-.5pt}) \theta^{\dagger}\!({\kern-.5pt}n{\kern-.5pt}) \! +\! \frac{1}{2} \frac{G(n)^{2} r}{\sqrt{n} \lambda \! +\! \sqrt{2r\! +\!\lambda} \sqrt{r\! +\! n \lambda}} \frac{\sqrt{2r\! +\!\lambda}}{\sqrt{r\! +\! n \lambda}} \theta^{\dagger}\!({\kern-.5pt}n{\kern-.5pt}) \right. \\ && \left. +G(n)^{2} (p-\psi^{\dagger}-\theta^{\dagger}(n)) \right. \\ && \left. \left( \! \frac{2 \sqrt{n} \lambda \! +\! \sqrt{2r\! +\! \lambda} \sqrt{r\! +\! n\lambda} }{2 (\sqrt{n} \lambda \! +\! \sqrt{2r\! +\! \lambda} \sqrt{r\! +\! n\lambda}) } \! +\! \frac{\sqrt{2r+\lambda} n \lambda }{2 \sqrt{r\! +\! n \lambda} (\sqrt{n} \lambda \! +\! \sqrt{2r\! +\!\lambda} \sqrt{r\! +\! n\lambda}) }\! \right) \bigg) \right) \\ &=&\! -1, \end{array} $$

$$ \begin{array}{@{}rcl@{}} sgn\!\left( \!\frac{\partial n^{\dagger}}{\partial p}\!\right)\! &=&\! sgn\!\left( \frac{\partial F}{\partial p}\right)\\ &=&\! sgn\bigg((G(n)^{2}+2nG'(n)G(n))(p-\psi^{\dagger}-\theta^{\dagger}(n))\\ &&\left. + \bigg(\frac{1}{2} \sigma_{y} \frac{\sqrt{r}}{\sqrt{n}} \frac{\sqrt{2r+\lambda}}{\sqrt{r+n \lambda}} +2nG'(n)G(n)\theta^{\dagger}(n) \bigg) \right) \\ &=&\! +1. \end{array} $$

To determine the impact of a change in the synergy effect G(n) on the optimal team size n^‡, we add an infinitesimally small μ to the function G(n), that is, G(n)_Δ = G(n) + μ, and re-derive the optimality condition in the proof of Proposition 1. Next, we compare this condition with the original optimality condition in the proof of Proposition 1. Since the marginal costs C^′(n) remain the same, the difference between the marginal benefits determines the direction of the impact (the argumentation is analogous to the proof of Proposition 1):

$$ \begin{array}{@{}rcl@{}} &&sgn\left( \frac{\partial {CEP}^{\dagger}_{G(n)_{\Delta}}}{\partial n} - \displaystyle\frac{\partial {CEP}^{\dagger}_{G(n)}}{\partial n} \right)\\ &=& sgn\left( \underbrace{\bigg(\frac{1}{2}G(n)_{\Delta}+nG'(n)\bigg)G(n)_{\Delta}\bigg(p-\psi^{\dagger}-\theta^{\dagger}(n) \displaystyle\frac{G(n)^{2}}{G(n)_{\Delta}^{2}} \bigg)^{2}}_{\text{I}} \right. \\ &-& \underbrace{\bigg(\frac{1}{2}G(n)+nG'(n)\bigg)G(n)\bigg(p-\psi^{\dagger}-\theta^{\dagger}(n)\bigg)^{2}}_{\text{I'}} \\ &+& \bigg(p-\psi^{\dagger}-\theta^{\dagger}(n) \displaystyle\frac{G(n)^{2}}{G(n)_{\Delta}^{2}}\bigg)\\ &\cdot&\underbrace{\bigg(\frac{1}{2} \sigma_{y} \displaystyle\frac{\sqrt{r}}{\sqrt{n}}\frac{\sqrt{2r+\lambda}}{\sqrt{r+n \lambda}}+2nG'(n)G(n)_{\Delta}\theta^{\dagger}(n) \displaystyle\frac{G(n)^{2}}{G(n)_{\Delta}^{2}}\bigg)}_{\text{II}} \\ &-& \left. \underbrace{\bigg(p-\psi^{\dagger}-\theta^{\dagger}(n)\bigg)\bigg(\frac{1}{2} \sigma_{y} \frac{\sqrt{r}}{\sqrt{n}}\frac{\sqrt{2r+\lambda}}{\sqrt{r+n \lambda}}+2nG'(n)G(n)\theta^{\dagger}(n)\bigg)}_{\text{II'}} \right) \\ &=& +1. \end{array} $$

The only difference between parts I and I’ as well as between parts II and II’ lies in the use of the function G(n)_Δ instead of G(n). By assumption, G(n)_Δ > G(n). Furthermore, the factor $\frac {G(n)^{2}}{G(n)_{\Delta }^{2}}$ appears in parts I and II with $\frac {G(n)^{2}}{G(n)_{\Delta }^{2}} \leq 1$, such that parts I and II exceed their counterparts I’ and II’. The positive impact also holds for an infinitesimally small change in a multiplicatively linked factor; that is, G(n)_Δ = G(n) ⋅ γ. □

Proof Proof of Proposition 3.

The variable compensations depend on b^‡ and y for the workers as well as $\beta _{y}^{\dagger }$, y, $\beta _{x}^{\dagger }$, and x for the manager. Performance measure y and accounting income x unambiguously increase with team size; that is, $\frac {\partial y}{\partial n}>0$ and $\frac {\partial x}{\partial n}>0$. The manager’s bonus rate $\beta _{x}^{\dagger }$ is independent of the team size n; that is, $\frac {\partial \beta _{x}^{\dagger }}{\partial n}=0$. The first derivative of the worker’s bonus rate is

$$\frac{\partial b^{\dagger}}{\partial n}=\frac{\sigma_{y} \!\left( 4 n G^{\prime}(n) \left( n \lambda \sqrt{2 r + \lambda}+\!\sqrt{n} \lambda \sqrt{n \lambda + r}+r \sqrt{2 r + \lambda}\right) + r G(n) \sqrt{2 r + \lambda}\right)}{2 n^{3/2} \sqrt{r} G(n)^{3} \sqrt{n \lambda+r}}\!>\!0.$$

Given the assumptions for the underlying parameters, this expression is larger than zero. The manager’s short-term bonus rate is given by $\beta _{y}^{\dagger }=\frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}\cdot b^{\dagger }$. Since we have already demonstrated that $\frac {\partial b^{\dagger }}{\partial n}>0$, we focus on the term $\frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}$. Given the underlying assumptions for the parameter values, this expression increases in n. We thus conclude that the manager’s and the workers’ compensations both increase in n. Next, we demonstrate that this increase is more pronounced for managers than for workers. We show that this is already the case if we compare only the short-term bonuses, and we can then conclude that it will hold for the total bonus as well. We need to show that $\frac {\partial b^{\dagger }}{\partial n}<\frac {\partial \beta _{y}^{\dagger }}{\partial n}$. We define b = b(n), $\frac {\partial b}{\partial n}=b^{\prime }(n)$, $\frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}=\beta (n)$, and $\frac {\partial \frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}}{\partial n}=\beta ^{\prime }(n)$. Applying basic differentiation rules leads to

$$b^{\prime}(n)<\beta^{\prime}(n)\cdot b(n)+\beta(n)\cdot b^{\prime}(n) \text{ and}$$

$$b^{\prime}(n)(1-\beta(n))<\beta^{\prime}(n)\cdot b(n).$$

It can be shown that β(n) > 1 for n > 2, such that (1 − β(n)) < 0 for n > 2. Consequently, the left-hand side of the expression $b^{\prime }(n)(1-\beta (n))$ is negative for n > 2, while the right-hand side, $\beta ^{\prime }(n)\cdot b(n)$, is positive. Therefore we conclude that the manager’s short-term bonus increases faster than the workers’ bonus for teams with more than two members. □

Proof Proof of Proposition 4.

It has already been shown in the proof of Proposition 2 that the optimal team size n varies with changes in exogenous parameters.²⁸ We now demonstrate that the optimal bonus rates b and β_y for an endogenously chosen n vary with changes in exogenous parameters. Since $\beta _{y}=\frac {\sqrt {n}\sqrt {n\lambda +r}}{\sqrt {2r+\lambda }}\cdot b=\beta (n)\cdot b$, with β(n) > 1 for n > 2 (see Proof of Proposition 3), it is sufficient to derive the comparative statics for b. For any exogenous parameter p,σ_p, and σ_y denoted by h, we have

$$\frac{db}{dh}=\frac{\partial b}{\partial h}+\frac{\partial b}{\partial n}\cdot\frac{dn}{dh}.$$

We derive the following proof exemplarily for a change in productivity p. The proof for the other exogenous parameter is analogous. We have already shown in the proof of Proposition 2 that $\frac {\partial n}{\partial p}>0$. Moreover, $\frac {\partial b}{\partial p}=1>0$, which confirms the first part in Proposition 4. Next, we have to show that there exist cases in which

$$\frac{db}{dh}>\frac{dn}{dh};$$

that is,

$$\frac{\partial b}{\partial p}+\frac{\partial b}{\partial n}\cdot \frac{dn}{dp}>\frac{dn}{dp}.$$

Rearranging yields

$$\frac{\partial b}{\partial p}+\frac{dn}{dp}\left( \frac{\partial b}{\partial n}-1\right)>0.$$

Since $\frac {\partial b}{\partial p}$ and $\frac {dn}{dp}$ point in the same direction, the above condition is fulfilled, for example, if $\frac {\partial b}{\partial n}\geq 1$, confirming the second part in Proposition 4. □

Proof Proof of Corollary 1.

We have already shown in the proof of Proposition 3 that $\frac {\partial b}{\partial n}>0$. We now have to show that $\frac {\partial n}{\partial b}>0$ holds as well. We take the procedure described in the proof of Proposition 2 and use implicit differentiation, where F is given by $\displaystyle \frac {\partial CEP^{\dagger }}{\partial n}$ before optimizing with respect to the bonus rate b:

$$\frac{\partial F}{\partial b}\! =\!\frac{-\sigma_{y}\!\left( \!\frac{(r+2n\lambda)\sqrt{2r+\lambda}}{\sqrt{n}\sqrt{r+n\lambda}}+2\lambda\!\right) + 2\!\left( \!\sqrt{r}{\kern-.5pt}({\kern-.5pt}p\! -\! b{\kern-.5pt})\! +\! \sigma_{p}{\kern-.5pt}({\kern-.5pt}\lambda\! -\! \sqrt{\lambda}\sqrt{2r\! +\! \lambda})\!\right)\! G{\kern-.5pt}({\kern-.5pt}n{\kern-.5pt}){\kern-.5pt}({\kern-.5pt}G{\kern-.5pt}({\kern-.5pt}n{\kern-.5pt})\! +\! 2nG^{\prime} {\kern-.5pt}({\kern-.5pt}n{\kern-.5pt}){\kern-.5pt})}{2\sqrt{r}}$$

At the optimal b^‡, this expression is larger than zero, confirming that team size increases with incentives. □

Proof Proof of Corollary 2.

While this result is already illustrated in Table 1, the analytical proof of the corollary follows directly from the fact that $\beta _{y}^{\dagger }=\frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}\cdot b^{\dagger }$, where $\frac {\sqrt {n} \sqrt {n \lambda +r}}{\sqrt {2 r+\lambda }}>1$ for n > 2.²⁹ The manager’s short-term incentives are thus affected by a change in exogenous parameters via two channels: a direct effect, which is visible in b^‡, and an indirect effect, through the optimal team size, n^‡. Both factors are directionally equally affected by changes in exogenous parameters. □

Appendix B

Proof Proof of the result ofFu et al. (2016) for a risk-averse principal.

To derive the impact of risk on team size for a risk-averse principal, we use the same solution procedure as in the original version but with a change in the principal’s utility:³⁰

$$ \phi = \rho(N) A N^{x} b^{\tau-1} - B N^{x} b^{\tau} - \frac{1}{2} \gamma N S^{2} b^{2} - \frac{1}{2} \lambda (\rho(N) - N b)^{2} S^{2}. $$

The result for the comparative statics is determined by the following inequality:

$$ sgn\left( \frac{\partial N}{\partial S^{2}} \right)=sgn\left( \frac{\partial^{2} \phi}{\partial N \partial b} \frac{\partial^{2} \phi}{\partial b \partial S^{2}} - \frac{\partial^{2} \phi}{\partial N \partial S^{2}}\frac{\partial^{2} \phi}{\partial b^{2}} \right) > 0. $$

Inserting first $\frac {\partial ^{2} \phi }{\partial b \partial S^{2}}$ and $\frac {\partial ^{2} \phi }{\partial N \partial S^{2}}$, we obtain the equivalent to the original condition (A10)³¹. After inserting, rearranging, and simplifying the equivalents of (A14) and (A16), (A18), and (A21), the condition is reduced to the equivalent of (A22):

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\! B N^{x} b^{\tau} (x -\frac{1}{2} \tau ) -\frac{\lambda}{\gamma} (\rho(N) -b N) B N^{x} b^{\tau-1} (x -\frac{1}{2} \tau ) \\ &&\!\!\!\! + \frac{1}{2}\frac{\lambda}{\gamma} A \rho(N) N^{x} (\tau -1) b^{\tau - 2} (\rho(N) -b N) (1 -\frac{2 \rho^{\prime}(N)}{b} ) \\ &&\!\!\!\! +\frac{1}{2}\frac{\lambda}{\gamma} N {S^{2}}\! \left( \! b^{2} N \! \!\left( {\kern-.3pt}N \lambda\! +\! \gamma\! \right) \! +\! \lambda \rho({\kern-.3pt}N{\kern-.3pt}) \! \left( \rho{\kern-.3pt}({\kern-.3pt}N{\kern-.3pt}) \! -\! 2 b {\kern-.3pt} N{\kern-.3pt} \right) \! -\! 2 \rho^{\prime}\!(N) \! \left( N \lambda \! +\! \gamma \right) \! \left( \rho(N) \! - \! b N \right)\! \right) \! >\! 0. \end{array} $$

For this inequality to hold, a few straightforward conditions need to hold. Moreover, a condition remains that can be simplified by inserting A and B:

$$ \alpha > \alpha_{min}=\left( \frac{1}{2} \frac{\rho(N)}{b} \left( 1 - 2 \frac{\rho^{\prime}(N)}{b} + \frac{b}{\rho(N)}\right) + \frac{1}{\delta} \right)^{-1 }. $$

Since α_min increases in b, the condition holds for any b < b_max if it holds for b_max. Using the limit on $b=b_{max}=\frac {\rho (N)}{N}$, we can rearrange this condition to be

$$ \alpha > \left( \frac{1}{2} N \left( 1 - 2 \frac{\rho^{\prime}(N)}{\rho(N)} N + \frac{1}{N} \right) + \frac{1}{\delta} \right)^{-1}. $$

Since $1 - 2 \frac {\rho ^{\prime }(N)}{\rho (N)} N + \frac {1}{N} > 1$, the condition holds for reasonable values for N and α. Using the numerical example in the paper, that is, δ= 2.1, N_min= 5, and the lower bound $1 - 2 \frac {\rho ^{\prime }(N)}{\rho (N)} N + \frac {1}{N}\text {=}1$, the inequality reduces to α > 0.336, which holds for the value in the paper; that is, α= 0.5. □