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.

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Notes

  1. 1.

    See, for instance, Balakrishnan and Sivaramakrishnan (2002), Carlton and Perloff (2005), Kaplan (2006), and Pittman (2009).

  2. 2.

    Buildings would be another example of an asset where degradation appears to be primarily time-driven.

  3. 3.

    This finding dates back to Arrow (1964).

  4. 4.

    The initial investment then effectively constitutes a private good to the extent that the investment enables an overall amount of output (operating hours) to be delivered. In contrast, if asset degradation is time-driven, the initial capacity investment enables a given stream of future outputs and, in that sense, constitutes a public good.

  5. 5.

    The same informational assumption is maintained in the earlier studies of Reichelstein (2000), Dutta and Reichelstein (2005), Baldenius and Ziv (2003), Bareket and Mohnen (2007), Friedl (2005), Pfeiffer and Schneider (2007), Wei (2004), and Johnson (2010).

  6. 6.

    If \(t<T\), the investment history becomes \(\varvec{I}_{t}=(I_{t-1},I_{t-2},\ldots ,I_{0},0,\ldots ,0)\).

  7. 7.

    As argued below, this informational assumption can be relaxed.

  8. 8.

    Since \(BV_{t}=BV_{t-1}-D_{t}+v\cdot I_{t}\), there is no need to include the current asset values \(BV_{t}\) in the performance measure.

  9. 9.

    See, for example, Rogerson (1997).

  10. 10.

    See, for instance, Nezlobin (2012).

  11. 11.

    This depreciation rule is consistent with the treatment of the right-of-use assets in Type B leases proposed by the IASB in its Exposure Draft #ED/2013/6.

  12. 12.

    See, for instance, Ehrbar (1998) and Young and O’Byrne (2000). Rajan and Reichelstein (2009) examine investment incentives under alternative depreciation rules if residual income is used as a performance measure.

  13. 13.

    Another possible reason is that firms may view it as costly to deviate from GAAP in designing their internal performance measures. See Young and O’Byrne (2000), pp. 267–268.

  14. 14.

    This depreciation rule is also known as the units-of-production method. Proportional depreciation appears to be a natural way to meet the IAS 16 requirement that “the depreciation method used shall reflect the pattern in which the asset’s future economic benefits are expected to be consumed by the entity.” In our model, the pattern of revenues generated by an asset may differ from the pattern according to which its capacity is consumed. We note that the proportional depreciation rule reflects the schedule of capacity consumption and not that of revenue generation, and therefore it is consistent with the IASB’s Exposure Draft #ED/2012/5, which emphasizes that depreciation rules should reflect the temporal pattern of consumption benefits inherent in the asset and that revenue is generally not a valid proxy for consumption.

  15. 15.

    Corollary 1 stands in contrast to the findings in Nezlobin (2012), who shows in the context of an equity valuation model that financial statements prepared using straight-line depreciation do not provide sufficient information for valuation purposes when assets have one-hoss shay productivity. Replacement cost accounting is effectively the only accounting rule that aggregates information about a firm’s past investments so that investors can still precisely value the firm based on the most recent financial statements. This uniqueness result is based on the observation that, to estimate the firm’s value, investors need to have information about its latest economic profits as well as the replacement cost of its assets in place. In contrast, we show that goal congruence only requires the owner to observe the firm’s current economic profit. The replacement cost of assets in place can be viewed as the value of “pre-paid” future capital costs, and, while knowing this quantity is necessary for equity valuation purposes, it is not necessary for incentive purposes.

  16. 16.

    The coefficient on book values, \(u_{bv}^{*}\left( \hat{r}\right)\), is equal to zero only when \(\hat{r}=0\). In that case, \(u_{d}^{*}\left( 0\right) =-\frac{\varvec{x\cdot 1}}{\varvec{x\cdot \gamma }}\).

  17. 17.

    See Dutta and Reichelstein (2010) for further details. Baldenius et al. (2014) study managerial performance evaluation in a model where firms can be subject to excess capacity for multiple periods.

  18. 18.

    See Observation 2 below.

  19. 19.

    The problem formulation in (9) reflects that the entire capacity investment must be made upfront. Models with a single investment opportunity are commonly considered in the economics literature [see, for example, Chapters 5 and 6 in Dixit and Pindyck (2002)]. In most environments, the need for an upfront capital investment is likely to originate from an incompleteness in the market for capacity services. In particular upfront fixed costs associated with the installation of assets (incurred in addition to the variable cost of \(v\cdot K_{0}\)) or a minimum capacity size come to mind as reasons for capacity investments to be inherently lumpy. Conceivably, the owner has sufficient information about the class of possible environments \(\{R_{t}(\cdot )\}_{t=1}^{T}\) to know that investments are profitable within a certain range, but making multiple investments is impractical due to the the minimum capacity constraint.

  20. 20.

    There may be multiple maximizers of \(V_{0}\). Our notion of goal congruence implies that the manager would be indifferent among any of these maximizers.

  21. 21.

    This asset valuation rule is conceptually similar to Hotelling’s (1931) rule for the price path of an exhaustible resource. In Hotelling’s formulation, there is no upfront investment expenditure but instead the firm must incur a periodic cost to extract the resource from the ground.

  22. 22.

    The arguments underlying Proposition 2 are related to Step 1 of the proof of Proposition 5 in Baldenius and Reichelstein (2005). In their model, asset values (inventories) are, by construction, independent of past utilization rates, that is, independent of the past \(\{s_{t}\}\). While this specification is plausible in the context of inventory valuation, our context of capacity assets suggests that depreciation charges may depend on the history of past production levels.

  23. 23.

    See, for instance, Ehrbar (1998), Ehrbar and Stewart (1999), Stewart (1991), and Young and O’Byrne (2000).

  24. 24.

    Models with multiplicative shocks to revenue functions are common in the economic literature on investment under uncertainty: see, for instance, Dixit and Pindyck (2002), p. 359. While the assumption of a multiplicative shock obviously entails a loss of generality, there is no further restriction entailed in assuming that the expected value of \(\tilde{\epsilon }\) is equal to one. We also note that our result in Proposition 3 can be extended to settings with multiple permanent shocks, possibly a new one arriving in each period.

  25. 25.

    For example, electricity is frequently generated by a combination of baseload and peaktime power plants. Baseload power plants (e.g., coal-fired plants) are usually represented as “one-hoss shay” type assets whose practical capacity is available uniformly for 40 years (NETL 2007). They tend to produce cheaper electricity than gas-powered peaker plants. As the name suggests, peaker plants can be turned on and off at short notice, and their remaining capacity is frequently rated in terms of output produced, which is highly correlated with the number of times the power generating unit has been activated.

  26. 26.

    Our model here can also be interpreted as one of a single asset the capacity of which declines with time and usage. Specifically, let \(K_{t}\) denote the total remaining capacity of the asset at date \(t\):

    $$\begin{aligned} K_{t}=(T-t)\cdot K_{0}^{b}+K_{0}^{s}-\sum \limits _{\tau =1}^{t}q_{t}^{s}. \end{aligned}$$

    We then have \(K_{t+1}=K_{t}-K_{0}^{b}-q_{t+1}^{s}.\) Thus capacity declines by \(K_{0}^{b}\) units in each period, plus by another \(q_{t+1}^{s}\) units due to “abnormal” usage in periods when \(q_{t+1}^{s}>0\). Over the asset’s useful life, the total abnormal usage has to satisfy

    $$\begin{aligned} \sum \limits _{t=1}^{T}q_{t}^{s}\le K_{0}^{s}. \end{aligned}$$
  27. 27.

    If \(v_{b}/T>v_{s}\), our model essentially reduces to the one with usage-driven asset degradation. On the other hand, if \(v_{b}<v_{s}\), it will never be optimal to invest in surge capacity, and the hybrid model reduces to the one with time-driven asset degradation.

  28. 28.

    Like in Rogerson’s (1997) framework, the common knowledge of the cost of investment implies that the marginal revenue at the optimal investment level is known at the outset, though only the better informed agent is in a position to determine that investment level.

  29. 29.

    Note that it follows from Eq. (28) that \(v^{b}\) exceeds \(v^{s}\cdot \left| \mathcal {T}^{*}\right|\).

  30. 30.

    Poterba and Summers (1992) demonstrate that internal hurdle rates used for capital budgeting purposes tend to exceed a firm’s actual cost of capital. See also Christensen et al. (2002) , Baldenius et al. (2007), Pfeiffer and Schneider (2007) and Johnson et al. (2013) for studies that capture the link between the underlying agency problem and the optimal capital charge rate.

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Acknowledgments

We are grateful to Tim Baldenius, Madhav Rajan, and seminar participants at the NYU/Yale joint seminar for comments on an earlier version of this paper. We are also grateful to Paul Fischer (the editor) and an anonymous reviewer for their constructive suggestions.

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Correspondence to Alexander Nezlobin.

Appendix

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. \(\square\)

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Nezlobin, A., Reichelstein, S. & Wang, Y. Managerial performance evaluation for capacity investments. Rev Account Stud 20, 283–318 (2015). https://doi.org/10.1007/s11142-014-9303-x

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Keywords

  • Performance measurement
  • Managerial incentives
  • Accrual accounting

JEL Classification

  • M40
  • M41
  • D82