Managerial performance evaluation for capacity investments

Abstract

This paper examines the design of managerial performance measures based on accounting information. The owners of the firm seek to create goal congruence for a better informed manager who is to decide on capacity investments and subsequent production levels. Managerial incentives are shaped by the performance metric and the depreciation schedule for capacity assets. Earlier literature has made the distinction between capacity assets whose degradation is primarily usage-driven as opposed to time-driven. Our analysis also distinguishes between two plausible scenarios in which an inherent lumpiness in the efficient scale of investments necessitates one upfront investment as opposed to a sequence of incremental capacity additions over time. For each of the four resulting scenarios, we obtain a complete characterization of the entire class of goal congruent performance metrics and depreciation schedules. The final part of our analysis also explores goal congruence in settings where the decline of asset productivity is a function of both time and usage.

Appendix

Proof of proposition 1

We first note that if a performance evaluation system $\left( \varvec{u},\varvec{d}\right)$ is goal congruent, then $u_{I}=0$. This follows immediately by observing that the utility payoff for a manager who plans to stay with the firm for only one period is given by $\alpha _{0}\cdot u_{I}\cdot v\cdot I_{0}.$ We next demonstrate that

$$\begin{aligned} u_{d}\cdot \varvec{d}+u_{bv}\cdot \varvec{bv}=-\varvec{z}^{*}. \end{aligned}$$

Assume that this equality did not hold. Let $\hat{z}_{\tau }=-u_{d}\cdot d_{\tau }-u_{bv}\cdot bv_{\tau -1}$ and $\hat{\varvec{z}}=\left( \hat{z}_{1},\ldots,\hat{z}_{T}\right)$. Let $\tau$ be the first position in which the vectors $\hat{\varvec{z}}$ and $\varvec{z}^{*}$ differ:

$$\begin{aligned} \tau =\min \left\{ i|\hat{z}_{i}\ne z_{i}^{*}\right\} . \end{aligned}$$

(35)

Consider a manager who intends to leave after for period $\tau$ and whose utility payoff is $U_{t}\left( \varvec{I}_{t}\right) =\pi _{\tau }$. At date 0, this manager will maximize:

$$\begin{aligned} \max _{I_{0},\ldots,I_{\tau -1}}R_{\tau } \left( K_{\tau }\right) +u_{d}D_{\tau }+u_{bv}BV_{\tau -1}. \end{aligned}$$

(36)

The objective function in (36) can be expressed as:

$$\begin{aligned} R_{\tau }\left( K_{\tau }\right) +u_{d}D_{\tau }+u_{bv}BV_{\tau -1}=R_{\tau }\left( x_{1}\cdot I_{\tau -1}+\cdots+x_{\tau }\cdot I_{0}\right) -\hat{z}_{1}\cdot v\cdot I_{\tau -1}-\cdots-\hat{z}_{\tau }\cdot v\cdot I_{0}. \end{aligned}$$

If the performance evaluation system is goal congruent, this function is maximized at $I_{0}^{*},\ldots,I_{\tau }^{*}$, which implies that all partial derivatives with respect to $I_{t}$ must be equal to zero at $\varvec{I}_{t}^{*}$. Therefore,

$$\begin{aligned} x_{\tau }R_{\tau }^{\prime }\left( x_{1}\cdot I_{\tau -1}^{*}+\cdots+x_{\tau }\cdot I_{0}^{*}\right) -\hat{z}_{\tau }\cdot v=0. \end{aligned}$$

Since $R_{\tau }^{\prime }\left( K_{\tau }^{*}\right) =c$, we obtain $x_{\tau }\cdot c-\hat{z}_{\tau }\cdot v=0$, and:

$$\begin{aligned} \hat{z}_{\tau }=\frac{x_{\tau }\cdot c}{v}=z_{\tau }^{*}, \end{aligned}$$

contradicting the definition of $\tau$ in Eq. (35). We have shown that

$$\begin{aligned} u_{d}\cdot \varvec{d}+u_{bv}\cdot \varvec{bv}=-\varvec{z}^{*}. \end{aligned}$$

(37)

We know that $bv_{0}=1$. Therefore the first components of the vectors in (37) determine $d_{1}$. Since $bv_{1}=bv_{0}-d_{1}$, we can solve for $d_{2}$ by referring to the second components of the vectors in (37). Proceeding iteratively, we obtain the unique solution to (37). It remains to show that it will coincide with $\varvec{d}^{*}\left( \hat{r}\right)$, where

$$\begin{aligned} \hat{r}=\frac{u_{bv}}{u_{d}}. \end{aligned}$$

To check that $\hat{r}\ne -1$, assume to the contrary that

$$\begin{aligned} u_{d}\left( \varvec{d}-\varvec{bv}\right) =-\varvec{z}^{*}. \end{aligned}$$

The last component of the vector in the left-hand side is zero, because any proper depreciation schedule is such that $d_{T}=bv_{T-1}$. However, $z_{T}^{*}\ne 0$ since $x_{T}\ne 0$; therefore, $\hat{r}$ cannot be equal to $-1$.

Denoting $\hat{\gamma }=\frac{1}{1+\hat{r}}$ and ${\hat{\varvec{\gamma}}}=\left( \hat{\gamma },\ldots,\hat{\gamma }^{T}\right)$, we observe that

$$\begin{aligned} \left( u_{d}\cdot \varvec{d}+u_{bv}\cdot \varvec{bv}\right) \cdot {\hat{\varvec{\gamma}}}=u_{d}\left( \varvec{d}+\hat{r} \cdot \varvec{bv}\right) \cdot {\hat{\varvec{\gamma}}}=u_{d}. \end{aligned}$$

On the other hand, Eq. (37) yields:

$$\begin{aligned} \left( u_{d}\cdot \varvec{d}+u_{bv}\cdot \varvec{bv}\right) \cdot {\hat{\varvec{\gamma}}}=-\varvec{z}^{*} \cdot {\hat{\varvec{\gamma}}}=-\frac{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}{{\varvec{x}} \cdot {\varvec{\gamma}}}. \end{aligned}$$

Therefore,

$$\begin{aligned} u_{d}=-\frac{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}{{\varvec{x}}\cdot {\varvec{\gamma}}} \quad {\text{and}}\quad u_{bv}=-{\hat{r}} \cdot \frac{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}{{\varvec{x}} \cdot {\varvec{\gamma}}}. \end{aligned}$$

Substituting these expressions back into (37), we obtain:

$$-\frac{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}{{\varvec{x}} \cdot {\varvec{\gamma}}} \cdot \varvec{d}-\hat{r}\cdot \frac{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}{{\varvec{x}} \cdot {\varvec{\gamma}}} \cdot \varvec{bv}=-\frac{{\varvec{x}}}{{\varvec{x}} \cdot {\varvec{\gamma}}},$$

or, equivalently,

$$\begin{aligned} \varvec{d}+\hat{r}\varvec{bv} = \frac{{\varvec{x}}}{{\varvec{x}} \cdot {\hat{\varvec{\gamma}}}}. \end{aligned}$$

(38)

Equation (38) has the unique solution, $\varvec{d=d}^{*}\left( \hat{r}\right)$. $\square$

Proof of proposition 2

It suffices to show that the compounded historical cost rule in conjunction with residual income would already emerge as the unique solution if the revenue functions were to belong to the following “test-class” of piece-wise linear functions:

$$R_{t} (q_{t} ) = \left\{ {\begin{array}{*{20}l} {q_{t} \cdot a_{t} ,} \hfill & {{\text{for}}} \hfill & {q_{t} \le \bar{q}_{t} ,} \hfill \\ {\bar{q}_{t} \cdot a_{t} + h \cdot \left( {q_{t} - \bar{q}_{t} } \right),} \hfill & {{\text{for}}} \hfill & {q_{t} \ge \bar{q}_{t} ,} \hfill \\ \end{array} } \right.$$

with $h<(1+r)^{t}\cdot v$. For this class of revenue functions, the optimal production levels satisfy $q_{t}^{*}=\bar{q}_{t}$ if $a_{t}>(1+r)^{t}\cdot v, q_{t}^{*}=0$ if $a_{t}<(1+r)^{t}\cdot v, q_{t}^{*}\in \left[ 0,\bar{q}_{t}\right]$ if $a_{t}=(1+r)^{t}\cdot v$. We first show that goal congruence requires that, whenever $a_{t}=(1+r)^{t}\cdot v$ for all $1\le t\le T$, it must be that $\pi _{t}=0$ for all $0\le t\le T$. To see this, suppose that to the contrary $\pi _{t}>0$ for some $t$. If the manager’s coefficient $\alpha _{t}$ on $\pi _{t}$ is positive, yet $\alpha _{\tau }=0$ for $\tau \ne t$, the agent would strictly prefer $q_{t}=\bar{q}_{t}$ to $q_{t}=0$. By continuity, $q_{t}^{*}=\bar{q}_{t}$ would also result in $\pi _{t}>0$ even if $a_{t}<(1+r)^{t}\cdot v$, which would violate goal congruence. A symmetric argument shows that $a_{t}=(1+r)^{t}\cdot v$ for all $t$ and $\pi _{i}<0$ for some $i$ would again result in a contradiction. We note that, $\pi _{0}=0$ implies $u_{I}=0$. This yields the following set of equations:

$$\begin{aligned} (1+r)^{t}\cdot v\cdot \bar{q}_{t}+u_{d}\cdot D_{t}+u_{bv}\cdot BV_{t-1}=0 \end{aligned}$$

for all $1\le t\le T$. Dividing by $v\cdot K_{0}^{*}=v\cdot \sum \bar{q}_{t}$, we obtain:

$$\begin{aligned} s_{t}\cdot (1+r)^{t}+u_{d}\cdot d_{t}+u_{bv}\left( 1-\sum \limits _{i=1}^{t-1}d_{i}\right) =0, \end{aligned}$$

(39)

for $1\le t\le T$. As $\left( \bar{q}_{1},..,\bar{q}_{T}\right)$ varies in $R^{T}$, these equations have to hold for all $(s_{1},\ldots,s_{T})$ on the unit simplex in $R^{T}$. Note that $d_{t}$ is a function of $s_{1},s_{2},\ldots s_{t}$ only. Recursive substitution for the Eq. in (39) shows that each $d_{t}$ is differentiable in its arguments. For $t=T$, (39) is equivalent to:

$$\begin{aligned} \left( 1-\sum \limits _{i=1}^{T-1}s_{i}\right) \cdot (1+r)^{T}+u_{d}\cdot \left( 1-\sum \limits _{i=1}^{T-1}d_{i}\right) +u_{bv}\cdot \left( 1-\sum \limits _{i=1}^{T-1}d_{i}\right) =0. \end{aligned}$$

(40)

Differentiating (40) with respect to $s_{T-1}$ and $s_{T-2}$ we obtain:

$$\begin{aligned} (u_{d}+u_{bv})\cdot \frac{\partial d_{T-1}}{\partial s_{T-1}}+(1+r)^{T}&= 0 \\ (u_{d}+u_{bv})\cdot \left( \frac{\partial d_{T-1}}{\partial s_{T-2}}+\frac{\partial d_{T-2}}{\partial s_{T-2}}\right) +(1+r)^{T}&= 0. \end{aligned}$$

At the same time, (39) implies:

$$\begin{aligned} \frac{\partial d_{T-1}}{\partial s_{T-2}}=\frac{u_{bv}}{u_{d}}\cdot \frac{\partial d_{T-2}}{\partial s_{T-2}}. \end{aligned}$$

Thus,

$$\begin{aligned} \left( 1+\frac{u_{bv}}{u_{d}}\right) \cdot \frac{\partial d_{T-2}}{\partial s_{T-2}}=\frac{\partial d_{T-1}}{\partial s_{T-1}}. \end{aligned}$$

On the other hand, (39) implies that

$$\begin{aligned} \left( 1+r\right) \cdot \frac{\partial d_{T-2}}{\partial s_{T-2}}=\frac{\partial d_{T-1}}{\partial s_{T-1}}. \end{aligned}$$

It follows that $\frac{u_{bv}}{u_{d}}=r$. Substitution into (39) yields:

$$\begin{aligned} s_{t}(1+r)^{t}+u_{d}[d_{t}+r\cdot bv_{t-1}]=0. \end{aligned}$$

(41)

Let

$$\begin{aligned} z_{t}\equiv d_{t}+r\cdot bv_{t-1}. \end{aligned}$$

By the Conservation Property of Residual Income, the present value of the $z_{t}$ is equal to one regardless of the depreciation schedule:

$$\begin{aligned} \sum \limits _{t=1}^{T}z_{t}\cdot \gamma ^{t}=1, \end{aligned}$$

if $\sum \nolimits _{t=1}^{T}d_{t}=1$ and $bv_{t}=1-\sum \nolimits _{i=1}^{t-1}d_{i}$. Since (41) is equivalent to $s_{t}(1+r)^{t}+u_{d}\cdot z_{t}=0$, the expression for the discounted sum becomes:

$$\begin{aligned} -u_{d}\cdot \sum \limits _{t=1}^{T}z_{t}\cdot \gamma ^{t} =\sum \limits _{t=1}^{T}s_{t}(1+r)^{t}\cdot \gamma ^{t}=\sum \limits _{t=1}^{T}s_{t}=1. \end{aligned}$$

Thus $u_{d}=-1$, and hence $u_{bv}=-r$. Finally, substituting these values for $u_{d}$ and $u_{bv}$ into (39), it follows that the depreciation schedule $\varvec{d}$ must coincide with the compounded historical cost rule as given in (14). $\square$

Proof of proposition 3

First, we consider the principal’s problem and show that the possibility of a unit-mean multiplicative shock to revenues does not alter the optimal investment and asset utilization plan. Let $(q_{1}^{*},q_{2}^{*},\ldots ,q_{T}^{*})$ denote the optimal asset utilization plan in a scenario with no shocks to revenues. When the principal anticipates a shock to revenues, he solves the following problem:

$$\begin{aligned} \max _{K_{0},\varvec{q}}\sum \limits _{t=1}^{\tau }R_{t}(q_{t}) \cdot \gamma ^{t}+E_{\epsilon }\left[ \sum \limits _{t=1}^{T-\tau }\tilde{\epsilon }\cdot R_{\tau +t}(q_{\tau +t}\left( \tilde{\epsilon }\right) )\cdot \gamma ^{t}\right] -v\cdot K_{0}, \end{aligned}$$

(42)

subject to

$$\begin{aligned} \sum \limits _{t=1}^{\tau }q_{t}+\sum \limits _{t=1}^{T-\tau }q_{\tau +t}\left( \epsilon \right) \le K_{0}, \quad \text { and }\quad q_{t}\ge 0,\quad q_{\tau +t}\left( \epsilon \right) \ge 0,\quad K_{0}\ge 0. \end{aligned}$$

Note that $q_{\tau +t}$ may, in principle, depend on $\epsilon$ for $t>0$. However, we will show that the optimal policy is still given by the vector $(q_{1}^{*},q_{2}^{*},\ldots ,q_{T}^{*})$ and does not depend on $\epsilon$.

Observe that for a given level of $K_{\tau }$ and a certain realization of $\epsilon$, the optimal production quantities $q_{\tau +t}\left( \epsilon \right)$ solve the problem

$$\begin{aligned} max_{\left\{ q_{\tau +t}\left( \epsilon \right) \right\} _{t=1}^{T-\tau }}\sum \limits _{t=1}^{T-\tau }\epsilon \cdot R_{\tau +t}(q_{\tau +t}\left( \epsilon \right) )\cdot \gamma ^{t} \end{aligned}$$

(43)

subject to

$$\begin{aligned} \sum \limits _{t=1}^{T-\tau }q_{\tau +t}\left( \epsilon \right) \le K_{\tau }. \end{aligned}$$

Clearly, the solution to this problem remains unchanged if one divides the objective function in (43) by $\epsilon$. Therefore the optimal $q_{\tau +t}$ is independent of $\epsilon$. The firm’s objective function (42) can be rewritten as

$$\begin{aligned} \sum \limits _{t=1}^{\tau }R_{t}(q_{t}) \cdot \gamma ^{t}&+ E_{\epsilon }\left[ \sum \limits _{t=1}^{T-\tau }\tilde{\epsilon }\cdot R_{\tau +t}(q_{\tau +t}) \cdot \gamma ^{t}\right] -v \cdot K_{0}=\sum \limits _{t=1}^{\tau }R_{t}(q_{t}) \cdot \gamma ^{t}+E_{\epsilon } \left[ \tilde{\epsilon }\right] \left( \sum \limits _{t=1}^{T-\tau }R_{\tau +t}(q_{\tau +t}) \gamma ^{t}\right) -v \cdot K_{0}. \end{aligned}$$

Since $E_{\epsilon }\left[ \tilde{\epsilon }\right] =1$, the firm’s objective function is the same as in the scenario with no shocks to revenues. Hence the optimal policy is to maximize $\gamma ^{t}\cdot R_{t}\left( q_{t}\right) -q_{t}\cdot v$ in all periods.

We next demonstrate that, if residual income based on the modified compounded historical cost rule is used as the performance measure, the manager will have an incentive to make optimal investment and production decisions. First, observe that, since $l_{\tau +t}$ do not depend on capacity utilization rates after date $\tau$, the manager’s objective function after date $\tau$ can be rewritten as:

$$\begin{aligned} \sum \limits _{i=\tau +1}^{T}\alpha _{i}\left( \epsilon \cdot R_{i}\left( q_{i}\right) -\epsilon \cdot d_{i}^{o}\cdot v\cdot K_{0}-r\cdot \epsilon \cdot bv_{i-1}^{o}\cdot K_{0}\right) +C\left( \epsilon \right) , \end{aligned}$$

(44)

where $C\left( \epsilon \right)$ is some constant that depends on $l_{\tau +1},\ldots,l_{T}$ but does not depend on $s_{\tau +1},\ldots,s_{T}$. From (44) it is clear that the manager’s choice of $q_{\tau +1},\ldots,q_{T}$ does not depend on $\epsilon$. Provided that $E_{\epsilon }\left[ l_{\tau +t}\right] =0$, the manager’s objective function at date 0 is equivalent to:

$$\begin{aligned}&\sum \limits _{i=1}^{\tau }\alpha _{i}\left( R_{i}\left( q_{i}\right) -d_{i}^{o} \cdot v \cdot K_{0}-r \cdot bv_{i-1}^{o} \cdot K_{0}\right) \\&\quad +E_{\epsilon }\left[ \sum \limits _{i=1}^{\tau }\alpha _{i}\left( \tilde{\epsilon } \cdot R_{i}\left( q_{i}\right) -\tilde{\epsilon }\;[d_{i}^{o} \cdot v \cdot K_{0}+r \cdot bv_{i-1}^{o} \cdot K_{0}]\right) \right] \\&\quad =\sum \limits _{i=1}^{\tau }\alpha _{i}\left( R_{i}\left( q_{i}\right) -d_{i}^{o}\cdot v\cdot K_{0}-r\cdot bv_{i-1}^{o}\cdot K_{0}\right) \\&\quad =\sum \limits _{i=1}^{\tau }\alpha _{i}\left( R_{i}\left( q_{i}\right) -\left( 1+r\right) ^{i}\cdot v\cdot q_{i}\right) . \end{aligned}$$

If the initial investment and all future production quantities are chosen according to the owner’s preferences, then every term in the summation above will be maximized, and so will be the manager’s utility. Therefore goal congruence is achieved.

Now assume that $\left( \varvec{u},\varvec{d}\right)$ is a goal congruent performance evaluation system. Consider a depreciation schedule $\varvec{d}'$ such that $d'_{t}=d_{t}$ for $t\le \tau$, and

$$\begin{aligned} d'_{t}\left( s_{1},\ldots,s_{t}\right) =E_{\epsilon }[d_{t} \left( s_{1},\ldots,s_{t},\epsilon \right) ]. \end{aligned}$$

We first argue that $\left( \varvec{u},\varvec{d}'\right)$ is a goal congruent performance evaluation system in the model with no shocks to revenues, that is, when revenue functions after period $\tau$ are given by $R_{\tau +t}\left( q_{\tau +t}\right) .$ Assume that $\left( \varvec{u},\varvec{d}'\right)$ is not goal congruent. Then there will exist a capacity level $K'_{0}$ and production quantities $q_{t}'$ such that

$$\begin{aligned} \sum \limits _{i=1}^{T}\alpha _{i}\left( R_{i}\left( q'_{i}\right) +u_{d}\cdot d_{i}^{'}\cdot v\cdot K_{0}^{'}+u_{bv}\cdot bv'_{i-1}\cdot v\cdot K_{0}^{'}\right) \\ \ge \sum \limits _{i=1}^{T}\alpha _{i}\left( R_{i}\left( q{}_{i}^{*}\right) +u_{d}\cdot d_{i}^{'}\cdot v\cdot K_{0}^{*}+u_{bv}\cdot bv'_{i-1}\cdot v\cdot K_{0}^{*}\right) .\nonumber \end{aligned}$$

(45)

Now consider the original model with a shock to revenues in period $\tau$. The expression on the right-hand side of (45) is equal to the expected utility that the manager obtains when making optimal investment and production decisions under the performance evaluation system $\left( \varvec{u},\varvec{d}\right)$. The expression on the left-hand side is the expected utility if the manager makes an initial investment of $v\cdot K_{0}$ and then implements production quantities $q'_{i}$ irrespective of the realization of $\epsilon$. Inequality (45) contradicts the goal congruence of $\left( \varvec{u},\varvec{d}\right)$. Therefore $\left( \varvec{u},\varvec{d}'\right)$ must be goal congruent in the model with no shocks to revenues.

By Proposition 2, goal congruence of $\left( \varvec{u},\varvec{d}'\right)$ implies that (1) the performance measure defined by $\varvec{u}$ is residual income and (2) $d_{t}'=d_{t}^{o}$. Therefore the first $\tau$ components of the vector $\varvec{d}$ are equal to the depreciation charges under the compounded historical cost rule. Let

$$\begin{aligned} l_{\tau +t}\left( s_{1},\ldots,s_{\tau +t},\epsilon \right) =d_{\tau +t}\left( s_{1},\ldots,s_{\tau +t},\epsilon \right) -\epsilon \cdot d_{\tau +t}^{o}\left( s_{1},\ldots,s_{\tau +t}\right) . \end{aligned}$$

Then,

$$\begin{aligned} E_{\epsilon }\left[ l_{\tau +t} \left( s_{1},\ldots,s_{\tau +t},\tilde{\epsilon }\right) \right] =d'_{\tau +t}-d_{\tau +t}^{o}=0. \end{aligned}$$

It can be immediately verified that $\sum \nolimits _{i=\tau +1}^{T}l_{i}=\left( 1-\epsilon \right) bv_{\tau }^{o}$. It remains to show that $l_{\tau +t}\left( s_{1},\ldots,s_{\tau +t},\epsilon \right)$ does not depend on $s_{\tau +1},\ldots,s_{\tau +t}$.

Let $l_{\tau +t_{0}}$ be the first one the depends on the asset utilization rates after date $\tau$. Then there will exist $s_{1},\ldots,s_{T}$ and $s'_{\tau +1},\ldots,s'_{T}$ such that $\sum \nolimits _{i=1}^{T}s_{i}=1, \sum \nolimits _{i=1}^{\tau }s_{i}+\sum \nolimits _{i=\tau +1}^{T}s'_{i}=1$, and for some $\epsilon _{0}$

$$\begin{aligned} l_{\tau +t_{0}}\left( s_{1},\ldots,s_{\tau +t_{0}},\epsilon _{0}\right) \ne l_{\tau +t_{0}}\left( s_{1},\ldots,s_{\tau },s'_{\tau +1},\ldots,s'_{\tau +t_{0}},\epsilon _{0} \right) . \end{aligned}$$

Fix some $K_{0}$ and consider the following revenue functions. For $t<\tau$,

$$R_{t} (q_{t} ) = \left\{ {\begin{array}{*{20}l} {q_{t} \cdot a_{t} ,} \hfill & {{\text{for}}} \hfill & {q_{t} \le \bar{q}_{t} ,} \hfill \\ {\bar{q}_{t} \cdot a_{t} + h \cdot q_{t} ,} \hfill & {{\text{for}}} \hfill & {q_{t} \ge \bar{q}_{t} ,} \hfill \\ \end{array} } \right.$$

where $\bar{q}_{t}=v\cdot K_{0},\,a_{t}>\left( 1+r\right) ^{t}\cdot v$ and $h<\left( 1+r\right) ^{t}\cdot v$. For $t>\tau$,

$$R_{t} (q_{t} ) = \left\{ {\begin{array}{*{20}l} {q_{t} \cdot a_{t} ,} \hfill & {{\text{for}}} \hfill & {q_{t} \le \hat{q}_{t} ^{{(1)}} ,} \hfill \\ {\hat{q}_{t} ^{{(1)}} a_{t} + h_{t} \cdot q_{t} ,} \hfill & {{\text{for}}} \hfill & {\hat{q}_{t} ^{{(1)}} \le q_{t} \le \hat{q}_{t} ^{{(2)}} ,} \hfill \\ {\hat{q}_{t} ^{{(1)}} a_{t} + h_{t} \cdot \hat{q}_{t} ^{{(1)}} + g_{t} \cdot q_{t} ,} \hfill & {{\text{for}}} \hfill & {q_{t} \ge \hat{q}_{t} ^{{(2)}} ,} \hfill \\ \end{array} } \right.$$

where $\hat{q_{t}}^{(1)}=\min \left\{ s_{t},s'_{t}\right\} K_{0},\,\hat{q_{t}}^{(2)} =\max \left\{ s_{t},s'_{t}\right\} K_{0},\,a_{t}>\left( 1+r\right) ^{t}\cdot v,\,h_{t}=\left( 1+r\right) ^{t}\cdot v, g_{t}<\left( 1+r\right) ^{t}\cdot v$. Given these revenue curves, the firm’s value will be maximized by investing $v\cdot K_{0}$ at date $0$ and then producing either according to the vector $\varvec{s}=\left( s_{1},\ldots,s_{T}\right)$ or to the vector $\varvec{s}'=\left( s_{1},\ldots,s_{\tau },s'_{\tau +1},\ldots,s'_{T}\right)$. By an argument similar to that in the proof of Proposition 2, it can be shown that goal congruence implies that for any realization of $\epsilon$, the manager’s performance measure in every period must be the same under the two production policies $\varvec{s}$ and $\varvec{s}'$. However, it is readily verified that residual income in period $\tau +t_{0}$ differs by $l_{\tau +t_{0}}\left(s_{1},\ldots,s_{\tau +t_{0}},\epsilon _{0}\right) -l_{\tau +t_{0}}\left( s_{1},\ldots,s_{\tau },s'_{\tau +1},\ldots,s'_{\tau +t_{0}},\epsilon _{0}\right)$ under the two production policies if $\epsilon _{0}$ is realized. Therefore a manager who only attaches a positive weight only to the performance measure in period $\tau +t_{0}$ will strictly prefer $\varvec{s}'$ to $\varvec{s}$ if $\epsilon _{0}$ is realized, which would contradict the assumed goal congruence of $(\varvec{u},\varvec{d})$. $\square$

Proof of lemma 1

Assume there exist two pairs, $\left\{ \mathcal {T}',MR'\right\}$ and $\left\{ \mathcal {T}'',MR''\right\}$, satisfying conditions (28), (29), and (30). If $MR'=MR''$, (29) and (30) imply that $\mathcal {T}'=\mathcal {T}''$. Without loss of generality, assume that $MR'<MR''$. Then, it follows from Eqs. (29) and (30) that $\mathcal {T}'\subset \mathcal {T}''$.

The following inequalities then lead to a contradiction:

$$\begin{aligned} v_{b}&= \sum \limits _{i\in \mathcal {T}^{''}}v_{s}+\sum \limits _{i\notin \mathcal {T}^{''}} \gamma ^{i}\cdot \beta _{i}\cdot MR^{''}\\&= \sum \limits _{i\in \mathcal {T}^{'}}v_{s}+\sum \limits _{i\in \mathcal {T}^{''} \setminus \mathcal {T}'}v_{s}+\sum \limits _{i\notin \mathcal {T}^{''}} \gamma ^{i}\cdot \beta _{i}\cdot MR^{''}\\&< \sum \limits _{i\in \mathcal {T}^{'}}v_{s}+\sum \limits _{i\notin \mathcal {T}^{'}} \gamma ^{i}\cdot \beta _{i}\cdot MR^{'}=v_{b}. \end{aligned}$$

$\square$

Proof of proposition 4

Our results in Sect. 3 imply that $H_{t}^{s}=\left( 1+r\right) ^{t}\cdot v^{s}\cdot q_{t}^{s}$ and

$$\begin{aligned} H_{t}^{bl}=\left( 1+r\right) ^{t}\cdot v^{s}\cdot K_{0}^{b}\cdot \mathcal {I}\left\{ t\in \mathcal {T}^{*}\right\} . \end{aligned}$$

Let us now verify that the depreciation rule specified in (33) satisfies the clean surplus condition. We have:

$$\begin{aligned} \sum \limits _{\tau =1}^{T}\gamma ^{\tau }\cdot z_{\tau }^{bl}=\sum \limits _{\tau \notin \mathcal {T}^{*}}^{T} \gamma ^{\tau }\cdot \frac{\beta _{\tau }}{\sum \nolimits _{i\notin \mathcal {T}^{*}}^{T}\gamma ^{i}\cdot \beta _{i}}=1, \end{aligned}$$

which is equivalent to the condition that the sum of corresponding depreciation charges is equal to one (Rogerson 1997).

The historical cost of base capacity in low demand periods is given by:

$$\begin{aligned} H_{t}^{bl}=\left( v^{b}-v^{s}\cdot \left| \mathcal {T}^{*}\right| \right) \cdot \frac{\beta _{t}}{\sum \nolimits _{i\notin \mathcal {T}^{*}}\gamma ^{i} \cdot \beta _{i}}\cdot K_{0}^{b}=\beta _{t}\cdot MR^{*}\cdot I_{b}, \end{aligned}$$

where the last inequality follows from (28).

Therefore residual income in high demand periods is given by:

$$\begin{aligned} RI_{t}=\beta _{t}\cdot R\left( K_{0}^{b}+q_{t}^{s}\right) -\left( 1+r\right) ^{t}\cdot v^{s}\cdot \left( K_{0}^{b}+q_{t}^{s}\right) . \end{aligned}$$

It follows from Eq. (25) that $K_{0}^{*b}$ and $q_{t}^{*s}$ maximize the expression above.

In low demand periods, residual income takes the following form:

$$\begin{aligned} RI_{t}=\beta _{t}\cdot R\left( K_{0}^{b}+q_{t}^{s}\right) -\left( 1+r\right) ^{t}\cdot v^{s}\cdot q_{t}^{s}-\beta _{t}\cdot MR^{*}\cdot K_{0}^{b}. \end{aligned}$$

Since $\beta _{t}\cdot MR^{*}\le \left( 1+r\right) ^{t}\cdot v^{s}$ for $t\notin \mathcal {T}^{*}$, it follows that $K_{0}^{b}=K_{0}^{*b}$ and $q_{t}^{s}=0$ maximize residual income in low demand periods. Therefore the optimal investment and capacity utilization decisions maximize residual income in all periods. $\square$