Managerial Compensation, Investment Decisions, and Truthfully Reporting

Abstract

This paper provides a formal analysis of investment decisions with special emphasis to mechanisms which induce managers to reveal their knowledge truthfully. In a one-period context ‘knowledge’ usually means the profit ratio. In a multi-period setting ‘knowledge’ is referred to the (multivariate) cash flow stream or the (univariate) net present value. Both situations are analysed in the paper. We start with the basic case ‘one firm, one manager’ and continue with the case ‘divisional firm, division managers’. With respect to the first case, we criticise two approaches (Rogerson, JPolE 105(4):770–795, 1997; Reichelstein, RAS 2(2):157–180, 1997) and develop a solution based on extended incentive contracts. To tackle the second case, we analyse pros and cons of Groves schemes.

Appendix

1.1 Proof of Theorem5.1

Proof

‘ ⇒ ’: Assume (2) holds true. Since this condition covers all possible cash flow streams, it also applies to non-stochastic streams x. Therefore, CE(x + z) needs to be independent of the specific value of z. In particular, it is allowed to determine z so that all components of x + z except the first one, x ₀ − z ₀, are set to zero, i.e. z _t = (−x _t+1 + z _t+1) ⋅ γ for all t = 0, 1, …, T − 1.

Following the recursion

$$\displaystyle{ \begin{array}{l@{\,\,}c@{\,\,}l@{\,\,}c@{\,\,}l} z_{T-1}\,\,& =\,\,& - x_{T}\cdot \gamma \,\, \\ z_{T-2}\,\,& =\,\,& - (-x_{T-1} + z_{T-1})\cdot \gamma \,\,& =\,\,& - x_{T-1} \cdot \gamma -x_{T} \cdot \gamma \,^{2} \\ z_{T-3}\,\,& =\,\,& - (-x_{T-2} + z_{T-2})\cdot \gamma \,\,& =\,\,& - x_{T-2} \cdot \gamma -x_{T-1} \cdot \gamma \,^{2} - x_{T} \cdot \gamma \,^{3}\\ \,\, & \,\, &\ldots \\ z_{0} \,\,& =\,\,& - (-x_{1} + z_{1})\cdot \gamma \,\,& =\,\,& - x_{1} \cdot \gamma -x_{2} \cdot \gamma \,^{2}\cdots - x_{T} \cdot \gamma \,^{T} \end{array} }$$

(37)

one arrives at x ₀ − z ₀ = NPV (x). Therefore, the certainty equivalents CE(x + z) and CE(NPV (x), 0, …, 0) coincide. Assuming (2) holds true, this implies CE(x) = CE(NPV (x), 0, …, 0), stating the indifference x ∼ (NPV (x), 0, …, 0). Since x is non-stochastic, the structure asserted in (3) follows immediately.

‘ ⇐ ’: Assume the structure asserted in (3) applies to the multiattributive utility function . Then, the date 0 certainty equivalent CE(c + z) is implicitly given by

$$\displaystyle{ \begin{array}{l@{\,\,}c@{\,\,}l} u(\mathit{CE}(\mathbf{c} + \mathbf{z}),0,\ldots,0)\,\,& =\,\,&\mathrm{E}u(\mathbf{c} + \mathbf{z}) =\mathrm{ E}u(\mathit{NPV }(\mathbf{c} + \mathbf{z}),0,\ldots,0) \\ \,\,& =\,\,&\mathrm{E}u(\mathit{NPV }(\mathbf{c}) + \mathit{NPV }(\mathbf{z}),0,\ldots,0)\,. \end{array} }$$

(38)

Since

$$\displaystyle\begin{array}{rcl} \mathit{NPV }(\mathbf{z})& =& -z_{0} + \frac{qz_{0} - z_{1}} {q} + \frac{qz_{1} - z_{2}} {q^{2}} +\ldots +\frac{qz_{T-2} - z_{T-1}} {q^{T-1}} + \frac{qz_{T-1}} {q^{T}} \\ & =& -z_{0} + z_{0} -\frac{z_{1}} {q} + \frac{z_{1}} {q} -\frac{z_{2}} {q^{2}} + \cdots + \frac{z_{T-2}} {q^{T-2}} -\frac{z_{T-1}} {q^{T-1}} + \frac{z_{T-1}} {q^{T-1}} = 0\;,\quad \quad {}\end{array}$$

(39)

CE(c + z) does not depend on z and, hence, condition (2) is fulfilled. □

1.2 Proof of Theorem5.2

Proof

Evaluating the shift in equity capital from period t − 1 to period t, EC _t − EC _t−1, one immediately arrives at

$$\displaystyle{ \mathit{EC}_{t} -\mathit{EC}_{t-1} = \mathit{NI}_{t} - c_{t}\;\Longleftrightarrow\;\mathit{NI}_{t} = c_{t} + \mathit{EC}_{t} -\mathit{EC}_{t-1}\,. }$$

(40)

Hence, the sum of discounted residual incomes can be written as

$$\displaystyle{ \begin{array}{@{}r@{\,\,}c@{\,\,}l@{}} \sum _{t=0}^{T}\mathit{RI}_{ t} \cdot \gamma \,^{t}\,\,& =\,\,&\sum _{ t=0}^{T}(c_{ t} + \mathit{EC}_{t} -\mathit{EC}_{t-1} - r \cdot \mathit{EC}_{t-1}) \cdot \gamma \,^{t} \\ \,\,& =\,\,&\sum _{t=0}^{T}c_{ t} \cdot \gamma \,^{t} +\sum _{ t=0}^{T}\mathit{EC}_{ t} \cdot \gamma \,^{t} -\sum _{ t=0}^{T}q\mathit{EC}_{ t-1} \cdot \gamma \,^{t} \\ \,\,& =\,\,&\sum _{t=0}^{T}c_{ t} \cdot \gamma \,^{t} +\sum _{ t=0}^{T}\mathit{EC}_{ t} \cdot \gamma \,^{t} -\sum _{ t=0}^{T}\mathit{EC}_{ t-1} \cdot \gamma \,^{t-1} \\ \,\,& =\,\,&\sum _{t=0}^{T}c_{ t} \cdot \gamma \,^{t} + \mathit{EC}_{ T} \cdot \gamma \,^{T}. \end{array} }$$

(41)

Taking RI ₀ = EC _T = 0 into account, (5) follows immediately. □

1.3 Proof of Property 5.1

Proof

Resolving the recursion, one arrives at

$$\displaystyle{ d_{t} =\zeta ^{-1}\cdot \bigg[c_{ t}+\sum _{j=1}^{t-1}c_{ j}\cdot \sum _{i=1}^{t-j}\Big(\begin{array}{@{}c@{}} t - j - 1 \\ i - 1 \end{array} \Big)\cdot r^{i}\bigg]-\sum _{ i=1}^{t}\Big(\begin{array}{@{}c@{}} t - 1 \\ i - 1 \end{array} \Big)\cdot r^{i}\,, }$$

(42)

where

$$\displaystyle{ \zeta =\sum _{ j=1}^{T}c_{ j} \cdot \gamma \,^{j}\,. }$$

(43)

Please note

$$\displaystyle{ \sum _{i=1}^{t-j}\Big(\begin{array}{@{}c@{}} t - j - 1 \\ i - 1 \end{array} \Big)\cdot r^{i} = r\cdot (1+r)^{t-j-1} = (1-\gamma )\cdot \gamma \,^{j-t} }$$

(44)

and

$$\displaystyle{ \sum _{i=1}^{t}\Big(\begin{array}{@{}c@{}} t - 1 \\ i - 1 \end{array} \Big)\cdot r^{i} = r\cdot (1+r)^{t-1} = (1-\gamma )\cdot \gamma \,^{-t}\,. }$$

(45)

Hence, (42) can be rewritten as follows

$$\displaystyle{ \begin{array}{@{}r@{\,\,}c@{\,\,}l@{}} d_{t}\,\,& =\,\,&\zeta ^{-1}\bigg[c_{ t} - (1-\gamma ) \cdot \gamma \,^{-t} \cdot \zeta +(1-\gamma ) \cdot \sum _{ j=1}^{t-1}c_{ j} \cdot \gamma \,^{j-t}\bigg] \\ \,\,& =\,\,&\zeta ^{-1}\bigg[\gamma \cdot c_{ t} - (1-\gamma ) \cdot \gamma \,^{-t} \cdot \zeta +(1-\gamma ) \cdot \sum _{ j=1}^{t}c_{ j} \cdot \gamma \,^{j-t}\bigg]\,. \end{array} }$$

(46)

We are now able to evaluate the sum

$$\displaystyle\begin{array}{rcl} \sum _{t=1}^{T}d_{ t}& =& \zeta ^{-1} \cdot \bigg [\gamma \cdot \sum _{ t=1}^{T}c_{ t} - (1-\gamma ) \cdot \zeta \cdot \sum _{t=1}^{T}\gamma ^{-t} + (1-\gamma ) \cdot \sum _{ t=1}^{T}\sum _{ j=1}^{t}c_{ j} \cdot \gamma \,^{j-t}\bigg] \\ & =& \zeta ^{-1} \cdot \bigg [\gamma \cdot \sum _{ t=1}^{T}c_{ t} + (1 -\gamma ^{-T}) \cdot \zeta +(1-\gamma ) \cdot \sum _{ t=1}^{T}\sum _{ j=1}^{t}c_{ j} \cdot \gamma \,^{j-t}\bigg] \\ & =& 1 +\zeta ^{-1} \cdot \bigg [\sum _{ t=1}^{T}c_{ t} \cdot (\gamma -\gamma \,^{t-T}) + (1-\gamma ) \cdot \sum _{ t=1}^{T}\sum _{ j=1}^{t}c_{ j} \cdot \gamma \,^{j-t}\bigg]\,. {}\end{array}$$

(47)

In order to prove d ₁ + … + d _T = 1 it is sufficient to verify that the term in square brackets is equal to zero. Since

$$\displaystyle{ \sum _{t=1}^{T}\sum _{ j=1}^{t}c_{ j} \cdot \gamma ^{j-t} =\sum _{ t=1}^{T}c_{ t} \cdot \sum _{j=0}^{T-t}\gamma ^{-j} =\sum _{ t=1}^{T}c_{ t} \cdot \frac{\gamma \,^{t-T}-\gamma } {1-\gamma } }$$

(48)

this is in fact the case. □

1.4 Proof of Theorem5.3

Proof

Since

$$\displaystyle\begin{array}{rcl} \sum _{t=1}^{\tau }\mathit{RI}_{ t} \cdot \gamma \,^{t}& =& \sum _{ t=1}^{\tau }\bigg[c_{ t} + d_{t} \cdot c_{0} + r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{ j}\bigg)\bigg] \cdot \gamma \,^{t} \\ & =& \sum _{t=1}^{\tau }\Bigg[c_{ t} +\mathop{\underbrace{ c_{0} \cdot \frac{c_{t}} {\sum \limits _{j=1}^{T}c_{j}\cdot \gamma ^{j}} - r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{j}\bigg)}}\limits _{=d_{t}\cdot c_{0}} + r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{ j}\bigg)\Bigg] \cdot \gamma \,^{t} \\ & =& \sum _{t=1}^{\tau }\Bigg[c_{ t} + c_{0} \cdot \frac{c_{t}} {\sum \limits _{j=1}^{T}c_{j} \cdot \gamma ^{j}}\Bigg] \cdot \gamma \,^{t} =\sum _{ t=1}^{\tau }\Bigg[1 + \frac{c_{0}} {\sum \limits _{j=1}^{T}c_{j} \cdot \gamma ^{j}}\Bigg] \cdot c_{t} \cdot \gamma \,^{t} \\ & =& \frac{1} {\sum \limits _{j=1}^{T}c_{j} \cdot \gamma ^{j}} \cdot \sum _{t=1}^{\tau }\Bigg[c_{ 0} +\sum _{ j=1}^{T}c_{ j} \cdot \gamma ^{j}\Bigg] \cdot c_{ t} \cdot \gamma \,^{t} = \frac{\sum \limits _{t=1}^{\tau }c_{ t} \cdot \gamma \,^{t}} {\sum \limits _{j=1}^{T}c_{j} \cdot \gamma ^{j}} \cdot \mathit{NPV }\,, {}\end{array}$$

(49)

manager’s present value of remuneration and the investment project’s NPV have the same sign as long as c ₁ > 0, …, c _T > 0. The latter is ensured when condition (18) is met. □

1.5 Proof of Theorem5.4

Proof

Since

$$\displaystyle\begin{array}{rcl} \mathrm{E}\bigg(\sum _{t=1}^{\tau }\mathit{RI}_{ t} \cdot \gamma \,^{t}\bigg)& =& \sum _{ t=1}^{\tau }\mathrm{E}(\mathit{RI}_{ t}) \cdot \gamma \,^{t} =\sum _{ t=1}^{\tau }\mathrm{E}\bigg[c_{ t} + d_{t} \cdot c_{0} + r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{ j}\bigg)\bigg] \cdot \gamma \,^{t} \\ & =& \sum _{t=1}^{\tau }\mathrm{E}\Bigg[c_{ t} +\mathop{\underbrace{ c_{0} \cdot \frac{\mathrm{E}(c_{t})} {\sum \limits _{j=1}^{T}\mathrm{E}(c_{j})\cdot \gamma ^{j}} - r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{j}\bigg)}}\limits _{=d_{t}\cdot c_{0}} + r \cdot c_{0} \cdot \bigg (1 -\sum _{j=1}^{t-1}d_{ j}\bigg)\Bigg] \cdot \gamma \,^{t} \\ & =& \sum _{t=1}^{\tau }\mathrm{E}\Bigg[c_{ t} + c_{0} \cdot \frac{\mathrm{E}(c_{t})} {\sum \limits _{j=1}^{T}\mathrm{E}(c_{j}) \cdot \gamma ^{j}}\Bigg] \cdot \gamma \,^{t} =\sum _{ t=1}^{\tau }\Bigg[1 + \frac{c_{0}} {\sum \limits _{j=1}^{T}\mathrm{E}(c_{j}) \cdot \gamma ^{j}}\Bigg] \cdot \mathrm{ E}(c_{t}) \cdot \gamma \,^{t} \\ & =& \frac{1} {\sum \limits _{j=1}^{T}\mathrm{E}(c_{j}) \cdot \gamma ^{j}} \cdot \sum _{t=1}^{\tau }\Bigg[c_{ 0} +\sum _{ j=1}^{T}\mathrm{E}(c_{ j}) \cdot \gamma ^{j}\Bigg] \cdot \mathrm{ E}(c_{ t}) \cdot \gamma \,^{t} \\ & =& \frac{\sum \limits _{t=1}^{\tau }\mathrm{E}(c_{t}) \cdot \gamma \,^{t}} {\sum \limits _{j=1}^{T}\mathrm{E}(c_{j}) \cdot \gamma ^{j}} \cdot \mathrm{ E}(\mathit{NPV })\,, {}\end{array}$$

(50)

manager’s expected present value of remuneration and the investment project’s expected NPV have the same sign as long as E(c ₁) > 0, …, E(c _T ) > 0. The latter is ensured when condition (20) is met. □