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Abstract

This paper identifies a novel effect which is crucial for the design of a management accounting information system. In contrast to prior literature, we explicitly model the firm's relationship to a supplier. We show that in addition to the previously identified trade-off – benefits of more information versus indirect or direct (agency) costs of information acquisition – another effect occurs: the input price effect. This effect influences the optimal design of the management accounting information system and changes the regimes where information acquisition is optimal for the principal. Also, in case of endogenous input prices we demonstrate that – perhaps surprisingly – paying an information rent to the agent can be beneficial because it works as a commitment towards an over-charging supplier to exploit the input price effect.

1. Introduction

Accounting research frequently studies the optimal design of a firm's information system which determines the allocation and the flow of information. Providing superior information to a self-interested manager might or might not be beneficial for the owner (e.g., Christensen, 1981 Christensen J. (1981). Communication in agencies. The Bell Journal of Economics, 12(2), 661674.[Crossref] [Google Scholar]). While better information improves decision-making and increases profits, private information can be used by the manager against the owner's best interest. To induce goal-congruent behavior, the owner must pay an information rent to the manager (e.g., Antle & Fellingham, 1995 Antle R., & Fellingham J. (1995). Information rents and preferences among information systems in a model of resource allocation. Journal of Accounting Research, 33, 4158.[Crossref], [Web of Science ®] [Google Scholar]). As a consequence, the owner might prefer a less informed agent by not installing an information system (Rajan & Saouma, 2006 Rajan M. V., & Saouma R. E. (2006). Optimal information asymmetry. The Accounting Review, 81(3), 677712.[Crossref], [Web of Science ®] [Google Scholar]). For example, if the owner designs the firm's budgeting mechanism, either top-down budgeting or participative budgeting might be beneficial (e.g., Baiman & Evans, 1983 Baiman S., & Evans J. H., III. (1983). Pre-decision information and participative management control systems. Journal of Accounting Research, 21(3), 371395.[Crossref] [Google Scholar]; Heinle, Ross, & Saouma, 2014 Heinle M. S., Ross N., & Saouma R. E. (2014). A theory of participative budgeting. The Accounting Review, 89(3), 10251050.[Crossref], [Web of Science ®] [Google Scholar]; Magee, 1980 Magee R. S. (1980). Equilibria in budget participation. Journal of Accounting Research, 18(2), 551573.[Crossref], [Web of Science ®] [Google Scholar]; Weiskirchner-Merten, 2019 Weiskirchner-Merten K. (2019). Interdependence, participation, and coordination in the budgeting process. Business Research, Advance online publication. Retrieved from https://doi.org/10.1007/s40685-019-0090-x. [Google Scholar]). Although the owner benefits from a manager's truthful report under participative budgeting, it is associated with slack building. Hence, top-down budgeting might be favored by the owner to avoid such real costs.

Research addresses this issue by focusing exclusively on the owner–manager relationship. The novelty of our paper is that we add a supplier who sells an input to the firm. We use this three-party setting to advance our understanding of the interplay between the choice of the firm's managerial accounting information system and the supplier's decision. Our economic model considers a firm that offers a final product in a monopolistic market. The owner makes the production decision and a manager implements production. To produce the final product, the firm purchases an input from a monopoly supplier. The supplier chooses a wholesale price and maximizes his expected profit. In addition to the input price, the firm faces stochastic marginal costs of production, which can be either low or high. Without an information system, production decisions must be based on expected marginal costs. If the owner installs an information system, the true marginal production costs are revealed only to the manager. The manager reports the costs to the owner and the manager receives a corresponding cost budget for production and a production schedule. To elicit a truthful report, the owner designs an incentive-compatible (direct) mechanism. Information about marginal costs enables the owner to ‘fine-tune’ the production schedule to the firm's true marginal costs (‘owner's value of true information’). At the same time, the owner must pay an information rent to the manager to elicit a truthful cost report (‘agent's information rent’).

To highlight the influence of the supplier, we first consider the equilibrium of our model for an exogenous input price. This benchmark case abstracts from any possible interplay between the firm's choice of installing an information system and the supplier's price response. In line with the literature, we find that if the owner's value of learning the true marginal costs dominates the agent's information rent, the owner prefers to install the information system and asks the manager for a cost report (and vice versa). The owner's decision is simple since providing superior private information for the manager is only profitable if the ‘value’ of information is high, i.e., if the spread between the realizations of marginal costs is sufficiently large.

We then assume that the firm purchases the input from a supplier. In contrast to the benchmark case of a fixed input price, we find that the owner's decision loses its simplicity since installing an information system can be optimal for a low as well as for a high spread between marginal costs. Intuitively, the owner's decision of installing an information system determines the firm's expected production quantity since the agent's information rent is reflected as an increase in production costs. A lower production quantity of the firm directly affects the supplier's expected profit and induces the supplier to decrease the input price. Hence, in addition to the two countervailing effects in the benchmark case, the owner benefits from this ‘input price effect’. The firm's commitment to install an information system and ask the manager for a cost report softens the supplier's pricing behavior. As a further consequence, even if the owner could install a perfect information system for free and learn the true marginal costs without involving the manager, the owner might still prefer to involve the manager and ask for a cost report to exploit this input price effect.

Our paper is most closely related to research on information system design where the principal controls the extent of information provided to the agent.11 In our setting, while the principal controls the information flow, information is exogenous. For work on endogenous information structure where the agent can expend costly effort to acquire information, see, e.g., Kessler (1998 Kessler A. S. (1998). The value of ignorance. The RAND Journal of Economics, 29(2), 339354.[Crossref], [Web of Science ®] [Google Scholar]), Hoppe (2013 Hoppe E. I. (2013). Observability of information acquisition in agency models. Economics Letters, 119(3), 104107.[Crossref] [Google Scholar]), Cremer, Khalil, and Rochet (1998 Cremer J., Khalil F., & Rochet J.-C. (1998). Contracts and productive information gathering. Games and Economic Behavior, 25, 174193.[Crossref], [Web of Science ®] [Google Scholar]). However, these papers do not consider the influence of an external party on optimal information provision.View all notes For example, Lewis and Sappington (1991 Lewis T. R., & Sappington D. E. M. (1991). All-or-nothing information control. Economics Letters, 37(2), 111113.[Crossref], [Web of Science ®] [Google Scholar]) analyze an adverse selection setting, where an owner can control the precision of the information the manager can learn. They show that it is optimal for the owner to either offer perfect (‘high-quality’) information or no (‘low-quality’) information. In Rajan and Saouma (2006 Rajan M. V., & Saouma R. E. (2006). Optimal information asymmetry. The Accounting Review, 81(3), 677712.[Crossref], [Web of Science ®] [Google Scholar]), the owner controls the flow of information to managers in the managerial accounting system. However, the owner lacks the expertise to interpret the accounting information. Consistent with Lewis and Sappington (1991 Lewis T. R., & Sappington D. E. M. (1991). All-or-nothing information control. Economics Letters, 37(2), 111113.[Crossref], [Web of Science ®] [Google Scholar]), they demonstrate that the owner always prefers an either perfectly informed or a perfectly uninformed manager.22 See Taylor and Xiao (2010 Taylor T. A., & Xiao W. (2010). Does a manufacturer benefit from selling to a better-forecasting retailer. Management Science, 56(9), 15841598.[Crossref], [Web of Science ®] [Google Scholar]) for a similar result in a manufacturer–retailer setting.View all notes If uncertainty is high, the benefit of having an informed manager exceeds the information rent paid to the manager (and vice versa). Rajan and Saouma (2006 Rajan M. V., & Saouma R. E. (2006). Optimal information asymmetry. The Accounting Review, 81(3), 677712.[Crossref], [Web of Science ®] [Google Scholar]) further study the optimal incentive intensity and the preferences of a manager regarding the choice of information system. Our paper contributes to this literature by studying the interplay between the firm's observable choice of information system and the decisions of an external third party. We identify a new effect, the input price effect, and show that the presence of the supplier has a crucial influence on the firm's choice of the information system.

Research on organizational design highlights that the optimal allocation of decision rights often balances the benefits from delegation against the control losses that are due to information asymmetries (e.g., Böckem & Schiller, 2014 Böckem S., & Schiller U. (2014). Managerial use of an information system. Working Paper, Retrieved from http://ssrn.com/abstract=1420160. [Google Scholar]). Arya, Glover, and Sivaramakrishnan (1997 Arya A., Glover J. C., & Sivaramakrishnan K. (1997). The interaction between decision and control problems and the value of information. The Accounting Review, 72(4), 561574.[Web of Science ®] [Google Scholar]) study information system design in a model of double moral hazard and address the decision-facilitating role and the decision-influencing (control) role of information. In line with our results, they show that the principal can be better off with less information.33 Likewise Cremer (1995 Cremer J. (1995). Arm's length relationships. The Quarterly Journal of Economics, 110(2), 275295.[Crossref], [Web of Science ®] [Google Scholar]2010 Cremer J. (2010). Arm's length relationships without moral hazard. Journal of the European Economic Association, Proceedings of the Twenty-Fourth Annual Congress of the European Economic Association, 8(23), 377387. [Google Scholar]) demonstrates that in agency settings lowering the costs of information acquisition can actually worsen the situation of the principal.View all notes However, in their model the main driver for this result is the interaction between the two roles of information such that less information serves as a commitment for the principal to reduce the bonus payments to the agent. In our setting, the firm is better off with less information since the supplier reacts to the firm's information problem by decreasing its input price. Arya, Fellingham, Glover, and Sivaramakrishnan (2000 Arya A., Fellingham J., Glover J. C., & Sivaramakrishnan K. (2000). Capital budgeting, the hold-up problem, and information system design. Management Science, 46(2), 205216.[Crossref], [Web of Science ®] [Google Scholar]) argue that managerial slack can work as a motivational device for the manager to spend more effort on the search for valuable projects. Therefore, the firm installs an information system that provides coarser and late information. Again, this paper focuses exclusively on players within the firm.

Finally, our paper is also related to work that studies the interplay between a firm's organizational design and the firm's input market (see, e.g., Arya & Mittendorf, 2010 Arya A., & Mittendorf B. (2010). Input markets and the strategic organization of the firm. Foundations and Trends in Accounting, 5(1), 197.[Crossref] [Google Scholar] for a survey). In our model, the firm faces an internal information problem and the economic consequences of double marginalization by the supplier who charges an input price above marginal costs. We show that paying the manager an information rent works as a pledge against an overcharging supplier.

The paper proceeds as follows. Section 2 provides the model setup. Section 3 studies the owner's choice of the firm's information system under the assumption of a fixed input price. Section 4 continues the analysis by considering a monopoly supplier. Section 5 discusses the robustness of our findings. Finally, Section 6 concludes. Proofs of the results are relegated to an Appendix, unless stated otherwise.

2. The Model

We consider the relationship between a firm owner (principal) and a manager (agent). The risk-neutral owner makes the production decision and the risk-neutral manager implements it. The firm serves a monopolistic output market. The demand for the product is described by the linear demand function p = ax, where p denotes the market price and x the market quantity of the final product. Owner and manager know the demand function. The firm's marginal production costs c are stochastic where c = 0 with probability φ and c = C with probability 1φ.44 Our results qualitatively hold for realizations c1>0 and c2>0. Furthermore, they also hold for three possible cost levels, e.g., c1=0,c2=C/2,c3=C with positive probabilities φ1,φ2,1φ1φ2. In Section 5, we provide more details and further describe the changes for the case where realizations of marginal costs are continuously distributed over an interval [0,C].View all notes Owner and manager (ex ante) share the same expectation about the marginal production costs. In what follows, we assume that a>C/(1φ), i.e., the highest willingness-to-pay in the market is sufficiently large compared to the adjusted marginal costs. This assumption avoids unnecessary case distinctions and guarantees that production quantities are positive.

To enable production, an input must be purchased from a monopolistic supplier. We assume that the supplier knows the demand function and the distribution of the firm's marginal costs. Given this information, the supplier charges a wholesale input price, w, to maximize its profit based on the firm's expected production quantity. For simplicity, we assume that each unit of the final product requires one unit of the input and w.l.o.g. we normalize the supplier's marginal costs to zero.

We study three scenarios. In the first scenario, the firm does not install an information system and information about the firm's marginal costs is symmetric for all players throughout the game. This scenario serves as a benchmark case for the other two scenarios where information about marginal cost is asymmetric. In the second information environment, the owner installs (at no additional costs) an information system which provides information about true marginal costs to the manager. The owner asks the manager to submit a report m about the true marginal costs. Based on the report m, the manager obtains a budget B(m) and is obliged to produce x(m) units. The manager might use private information to overstate marginal costs and consume resources not needed for production.55 This is in line with the general idea of slack in the literature captured by paying an information rent to the manager to induce truthful reporting of the manager's private information (e.g., Schiff & Lewin, 1970 Schiff M., & Lewin A. Y. (1970). Impact of people on budgets. The Accounting Review, 45(2), 259268.[Web of Science ®] [Google Scholar]).View all notes The third information environment assumes that the firm owner installs a perfect information system which reveals the true marginal costs to the owner at costs K without involving the manager. In this case, the owner saves the information rent paid to the manager but has to burden the costs K of perfect information. Summarizing, we consider one scenario with no additional information and two scenarios in which obtaining information is (directly or indirectly) costly. In the subsequent analysis, we will focus on the owner's design of an optimal (truth-telling) mechanism and we will derive conditions (on the marginal costs C) such that either information system is preferred.

The timing of decisions is as follows (see Figure 1). First, the owner decides whether to install an information system or not. This decision is observed by the supplier. Second, the supplier anticipates the firm's cost of production (including possible agency costs) and calculates the firm's expected quantity. Based on this prediction, the supplier determines the corresponding profit-maximizing input price w. Third, the owner asks the manager for a report and fixes the production budget and the associated production quantity. The manager receives the cost budget for production based on the budgeted production costs per unit. Finally, the production quantity is produced and sold in the market and profits are realized.

Figure 1. Timeline.

A key assumption of our analysis is that the transaction between the firm and its supplier is governed by a linear contract. If the supplier could, e.g., use a two-part tariff involving a unit price equal to marginal cost and a fixed fee to extract the entire surplus, then input market considerations seem to be moot (see also the discussion in Mittendorf, Shin, & Yoon, 2013 Mittendorf B., Shin J., & Yoon D.-H. (2013). Manufacturer marketing initiatives and retailer information sharing. Quantitative Marketing and Economics, 11, 263287.[Crossref], [Web of Science ®] [Google Scholar], p. 282). However, Arya and Mittendorf (2010 Arya A., & Mittendorf B. (2010). Input markets and the strategic organization of the firm. Foundations and Trends in Accounting, 5(1), 197.[Crossref] [Google Scholar], p. 79) provide various reasons why under nonlinear pricing arrangements the input price might not be equal to marginal costs. Hence, the assumption that contractual imperfections govern firm–supplier relationships seems reasonable.

A further critical assumption in our model is that the supplier is able to observe the owner's choice of information system, but not the details of the communication process that governs the interaction between the owner and manager. For example, based on a firm's reputation, the market might obtain a signal about the firm's internal controls or corporate governance structure but commonly external parties do not have access to detailed information about internal communications. This assumption is in line with the criticism raised about strategic transfer prices where observability of the chosen transfer prices is key. It has been argued (e.g., Göx, 2000 Göx R. F. (2000). Strategic transfer pricing, absorption costing, and observability. Management Accounting Research, 11, 327348.[Crossref] [Google Scholar]) that this issue can be resolved if the choice of the method of setting transfer prices (full cost versus marginal cost) can be observed instead of the exact values of the transfer prices. To summarize, suppliers must be privy to the internal controls and said controls must be sticky or costly to change.

3. Equilibrium with Exogenous Input Price

In this section, we study a situation where the price of the input, w, is exogenously fixed. Intuitively, this situation arises if the input is made in-house by the firm at marginal production costs of w or if the supplier market is perfectly competitive and input is supplied at a constant market price of w. Put differently, we abstract from any double marginalization issue between the firm and the monopoly supplier.

3.1. No Information System

If no information system is installed, the owner has to make an uninformed decision, i.e., the production quantity has to be based on expected costs, (1φ)C. The firm solves the following unconstrained optimization problem: maxx Π=(1φ)(aCwx)x+φ(awx)x.

The solution to this problem is xn=12(awC(1φ)) and the firm's expected profit is Πn=xn2 as long as awC(1φ)0 and both are 0 otherwise.66 The subscript n denotes the information scenario where no player has information about the true marginal costs.View all notes It is important to note that xn depends on the exogenous input price w and xn=E[xn], i.e., the quantity xn is deterministic since production can be based only on the expected marginal costs.

3.2. Firm Installs Information System

If the owner installs an information system, the manager is privately informed about the true marginal costs. The owner asks the manager for a report (m) about the observed marginal costs. To elicit the manager's private information, the owner can design a menu of contracts that specifies a production budget B(m) and an associated quantity x(m). According to the revelation principle, the owner can restrict attention to mechanisms that induce truth-telling, i.e., to mechanisms that induce the manager to report the true marginal costs c = 0 or c = C. Given that the manager reports truthfully, the firm's production quantity can be conditioned on the true marginal costs. However, to induce the manager to report the observed marginal costs truthfully, the owner has to pay an information rent to the manager. The principal's mechanism design problem can be written as (1) maxB(),x()Π=φ((ax(0)w)x(0)B(0))+(1φ)((ax(C)w)x(C)B(C))(1) subject to (PC1)B(C)Cx(C)0(PC2)B(0)0(IC1)B(C)Cx(C)B(0)Cx(0)(IC2)B(0)B(C). In addition to the manager's participation constraints (PC1) and (PC2), the owner's mechanism design problem has to include the incentive compatibility constraints (IC1) and (IC2) which ensure that the manager reports truthfully if the true marginal cost are C and 0 respectively.

From standard analysis, it follows that the participation constraint (PC1) and the incentive compatibility constraint (IC2) are binding. The remaining two inequalities can be ignored.77 See, for example, Laffont and Martimort (2002 Laffont J.-J., & Martimort D. (2002). The theory of incentives. the principal-Agent model. Princeton, NJ: Princeton University Press.[Crossref] [Google Scholar]).View all notes This leads to B(C)=Cx(C) and B(0)=B(C)=Cx(C). If the manager observes low marginal costs, c = 0, he receives an information rent since B(0)=Cx(C)>0. Inserting B(C) and B(0) into the principal's objective function yields the unconstrained problem (2) maxx()φ(ax(0)w)x(0)+(1φ)ax(C)wC1φx(C),(2) and solving leads to xi(0)=12(aw) and xi(C)=12(awC/(1φ)).88 The subscript i denotes the information scenario where the owner obtains information about the true marginal costs from the manager.View all notes Note that we have xi(0)=0 if and only if w>a and xi(C)=0 if and only if w>aC/(1φ).

Assuming that all quantities are positive, we have Πi=φxi(0)2+(1φ)xi(C)2. If xi(C)=0 (or equivalently w>aC/(1φ)), we obtain Πi=φxi(0)2.

Alternatively, the owner can install a perfect information system. In this case, the owner learns the true marginal costs at costs K without involving the manager. The owner's maximization problem is the same as in (2) with the difference that the expected marginal costs C/(1φ) are replaced by the true marginal costs C. It is easy to see that the optimal production quantities are xI(0)=12(aw) and xI(C)=12(awC).99 The subscript I denotes the information scenario where the owner obtains information about the true marginal costs by installing a perfect information system.View all notes Assuming that both production quantities are positive, the firm's expected profit is ΠI=φxI(0)2+(1φ)xI(C)2K. If the firm only produces in case of low costs (C>aw), we have ΠI=φxI(0)2K.

3.3. Comparison of the Information Regimes

A simple comparison of the expected profit without information, Πn, and the expected profit with information, Πi, yields conditions on C under which the firm prefers to obtain information about the true marginal production costs.

We start the analysis assuming w<aC/(1φ) such that the firm's quantity is positive even if marginal costs turn out to be high. Notice first that the owner's derived objective function in (2) is identical to the expected profit of an informed firm owner who faces marginal production costs of c = 0 with probability φ and c=C/(1φ) with probability 1φ. Denoting the random variable that captures these marginal production costs by cˆ, we conclude that Var(cˆ)=C2φ/(1φ). Then, the firm's expected profit Πi in the case where production is positive independent of the realization of marginal costs (see Section 3.2) can be decomposed as Πi=x¯n2+14Var(cˆ). Here, x¯n denotes the quantity an uninformed owner would choose given a cost structure that incorporates the agency costs. In other words, x¯n is the solution to the maximization problem maxx (axwC)x where the quantity cannot be conditioned on true marginal costs and expected production costs are E[cˆ]=C>C(1φ), i.e., larger than the (ex-ante) expected marginal production costs due to the information problem. The term 14Var(cˆ) arises from the possibility of the owner to condition his production decision on the true cost and is therefore equivalent to the expected value of perfect information (see, e.g., Hirshleifer & Riley, 1992 Hirshleifer J., & Riley J. G. (1992). The analytics of uncertainty and information. Cambridge: Cambridge University Press.[Crossref] [Google Scholar]). Using this decomposition, installing the information system is preferred by the owner if and only if x¯n2+14Var(cˆ)>xn2. This condition captures the trade-off the owner faces in the decision to install an information system. On the one hand, installing an information system enables the owner to condition the production decision on the true marginal costs and this benefit is captured by the expected value of perfect information (EVPI). On the other hand, the manager uses the information system to obtain private information and to learn the true costs, the owner has to pay an information rent to the manager. The situation is akin to a firm being confronted with increased expected production costs. Hence, the resulting quantity x¯n is strictly smaller than xn. However, since the EVPI is strictly increasing in C, the firm can be better off if C is sufficiently large. In fact, it can be shown that for C>Cb:=2(aw)(1φ)33φ+φ2 installing an information system is beneficial. The cut-off value Cb in the case of a fixed input price satisfies the assumption that the firm produces in both states if and only if φ12(35). Figure 2 illustrates the situation.

Figure 2. The figure shows the firm's expected profits Πn (solid curve) and Πi (dash-dotted curve) in the benchmark case as functions of C for a = 6, φ=0.2, and w = 1. Below Cb it is not optimal to install an information system and above Cb it is strictly better to install an information system. For the parameter values used here, we have Cb=2(1φ)(aw)/(φ3)(φ+3) because of φ12(35).

In the alternative case where the firm does not produce in case of high costs (xi(C)=0), i.e., if w>aC/(1φ), the firm's expected profit can be decomposed as Πi=x¯n2+EVPI, with EVPI:=14φ(aw)214(aCw)2. The expected value of perfect information is again increasing in C. Installing an information system is preferred by the owner if C>Cb=(aw)/(1+φ) and the cut-off value Cb satisfies the assumption that the firm only produces if marginal costs are low (i.e., Cb>(aw)(1φ)) if and only if φ>12(35).1010 In order to guarantee that Cb<Cmax=a(1φ), the condition (1φ)(1+φ)1>w/a has to be fulfilled.View all notes

The following proposition summarizes our results.

Proposition 3.1

Consider the benchmark case of a fixed input price. Let Cb=2(aw)(1φ)/(33φ+φ2) for φ12(35) and Cb=(aw)/(φ+1) for φ>12(35). The firm earns a higher profit if the owner installs an information system and asks the manager for a report if and only if C>Cb. For C<Cb, installing no information system is more profitable for the firm. For C = 0 or C=Cb, the owner is indifferent between installing an information system or not.

The first main take-away from our analysis of the case with a fixed input price is that the decision to install an information system is quite straightforward. The owner should install an information system if the marginal cost C (or the cost spread between the two states) is sufficiently large while no information system should be installed otherwise. In the case of a large marginal cost spread, the value of learning the true marginal costs is large and dominates the loss due to agency costs.

The second main take-away from our analysis concerns the owner's possibility to install a perfect information system. Assume that the owner can learn the true costs for free (K = 0) and, for simplicity, assume that all production quantities are positive. The firm's expected profits are given by Πn, Πi, and ΠI (see Sections 3.1 and 3.2). The case where the owner relies on ex-ante information is dominated by the case where the owner can obtain perfect information at no additional costs since the owner prefers to condition production quantities on true (and not expected) marginal costs. Likewise, the case where the owner installs an information system and asks the manager for a report is dominated by the case where the owner can obtain perfect information at no additional costs (K = 0) since the owner prefers to save the manager's information rent. The bottom line of these arguments is that if the input price is exogenous, then the owner always prefers to have perfect information if it comes at no costs.

For strictly positive values of K, a comparison of Πi and ΠI shows that it is better to install a perfect information system for intermediate values of C and to install an information system and ask the manager for a cost report for low and high values of C. Intuitively, for C = 0 there is no uncertainty and therefore both information systems are equivalent. For large values of C, there is no production in case of high costs for both information systems and, hence, expected profits are equal except for the costs K. Therefore, for low and high values of C the imperfect information system dominates Πi>ΠI. It is not difficult to show that ΠIΠi is single-peaked and, therefore, for medium values of C the perfect information system dominates as long as K is not too large. If K is too large, then obviously the perfect information system is too expensive and therefore never optimal.

4. Optimal Information in Presence of a Supplier

We now analyze the influence of a monopoly supplier on the owner's decision to install an information system. In line with the previous section, we first derive the firm's expected profit if no information system is installed and the firm has to rely on ex-ante information about its marginal production costs. We will build on the results obtained for an exogenous input price and additionally consider the endogenous price choice of a monopoly supplier. Then we analyze a situation where the owner installs an information system and asks the manager for a report about marginal costs. By comparing the expected profits, we can determine under which condition on C it is profitable for the owner to install an information system and how endogenous supplier pricing affects the conditions for optimality.

4.1. No Information System

The analysis of the case with a monopoly supplier is similar to the case with an exogenous supplier price with the difference that there is an additional stage where the monopoly supplier endogenously chooses its profit-maximizing input price. As in Section 3.1, backward induction yields the firm's quantity xn=12(awC(1φ)) and the firm's expected profit Πn=xn2=14(awC(1φ))2 as long as awC(1φ)0 and both are 0 otherwise. The quantity is always positive in equilibrium due to the assumption that a>C/(1φ). The monopoly supplier anticipates the firm's quantity choice xn and (given our assumption that the supplier's marginal costs are zero) determines the input price w such that Πns=wxn(w) is maximized. This yields the input price wn=12(a(1φ)C). Note that wn>0. We can now obtain the firm's and supplier's equilibrium profits by inserting the optimal input price wn and the firm's optimal quantity xn evaluated at w=wn into Πn and Πns. The following lemma summarizes our findings. The proof of the lemma is straightforward and is therefore omitted.

Lemma 4.1

If the owner does not install an information system, the quantity xn=(a(1φ)C)/4 is optimal. The supplier charges the price wn=(a(1φ)C)/2 for the input. The firm's equilibrium profit is Πn=((a(1φ)C)2)/16 and the supplier's equilibrium profit is Πns=(a(1φ)C)2/8.

Obviously, we have the identity Πn=Πn(w=wn), where Πn(w=wn) denotes the firm's expected profit with an exogenous input price (see Section 3.1) evaluated at the optimal input price wn given in lemma 4.1. We will use this identity later in our analysis.

4.2. Firm Installs Information System

Again, in the case with a monopoly supplier there is an additional stage where the monopoly supplier endogenously chooses its profit-maximizing input price. Therefore, as in Section 3.2, backward induction yields the firm's quantities xi(0)=12(aw) and xi(C)=12(awC/(1φ)), where xi(C)=0 if and only if w>aC/(1φ). The corresponding expected profit Πi of the firm is given in Section 3.2. The monopoly supplier anticipates the firm's quantity choices xi(0) and xi(C) and determines the input price w. Although the supplier can anticipate the quantities that are induced by the firm's truth-telling mechanism in case of low (c = 0) and high (c = C) marginal costs, since the input price is chosen before marginal costs are revealed the supplier is not able to condition the input price on the true marginal costs. Therefore, the supplier charges an input price based on the expected quantity and the supplier maximizes Πis=w(φxi(0)+(1φ)xi(C)). In detail, the supplier's profit is now given by1111 In the case w>a, the supplier's expected profit is zero. Such a high input price is never optimal, however, since the supplier could achieve a positive profit by setting an input price w<a. Therefore, we can restrict our analysis to the interval 0<w<a.View all notes (3) Πis=wφaw2+(1φ)awC/(1φ)20<waC1φ,wφaw2aC1φ<wa.(3) The optimal input price of the supplier can be determined by finding the globally optimal values of w for the two cases 0<waC/(1φ) and aC/(1φ)<wa. Note that in the first case the firm produces under high and low marginal costs and the resulting supplier's profit is given by the first line in (3). In the second case, the firm only produces under low marginal costs and the supplier's profit is given by the second line in (3). At the threshold value w=aC/(1φ), the two profits coincide and, therefore, the supplier's expected profit is continuous in w (but not differentiable). The following lemma gives the supplier's optimal input price as a function of C and the firm's optimal quantities and the expected profits of the parties. A proof of the first part of the lemma can be found in the mathematical appendix. The derivation of the second part is straightforward and is therefore omitted.1212 The firm's quantities follow from inserting the input price wi into xi(0)=12(aw) and xi(C)=12(awC/(1φ)), where xi(C)=0 for C>Cˆ. Using the input price and the firm's quantities, the expected profits can be derived.View all notes

Lemma 4.2

There exists a value Cˆ=a(1φ) such that the following results hold.

  1. For C<Cˆ the supplier charges wi=(aC)/2 and for C>Cˆ the supplier charges wi=a/2. In the boundary case C=Cˆ the supplier is indifferent between w=(aC)/2 and w=a/2.

  2. For C<Cˆ, the firm's quantities are xi(0)=(a+C)/4 and xi(C)=(a(1φ)C(1+φ))/4(1φ). The resulting expected profit of the firm is Πi=((aC)2φ(a3C)(a+C))/16(1φ) and the expected profit of the supplier is Πis=(aC)2/8.

  3. For C>Cˆ, the firm's quantities are xi(0)=a/4 and xi(C)=0. The resulting expected profit of the firm is Πi=φa2/16 and the expected profit of the supplier is Πis=φa2/8.

Figure 3 further illustrates part (i) of the lemma. The figure depicts the supplier's expected profit as a function of C for the case where the firm produces a positive quantity even if c = C and for the case where the firm stops production if marginal costs are high. It is shown that the supplier's input price increases from wi=(aC)/2 to wi=a/2 before the firm's quantity in case of high costs becomes zero.

Figure 3. The figure shows the supplier's expected profit as a function of C for a = 6 and φ=0.3. The solid curve represents the expected profit Πis=18(aC)2 given that the firm produces a positive quantity in both states. The optimal input price in this case is wi=(aC)/2. The dash-dotted horizontal line represents the expected profit Πis=φ(a2/8) (Πis(a/2)) given that the firm does not produce in case of high costs. In this case, wi=a/2. At Cˆ where the two curves intersect, the supplier's optimal input price jumps upwards from w=(aC)/2 to w=a/2. The dashed line depicts the firm's quantity xi in case of high costs given an input price of wi=(aC)/2. The figure reveals that the jump in the input price at Cˆ occurs in an admissible region (i.e., before the firm's quantity in case of high costs becomes 0).

Obviously, due to the backward induction procedure we have the identity Πi=Πi(wi=wi) where for C<Cˆ the expression Πi(wi=wi) denotes the firm's expected profit Πi where production is positive for high and low marginal costs evaluated at the input price wi=(aC)/2. For C>Cˆ, the expected profit Πi=φxi(0)2 and wi=a/2 have to be used.

4.3. Comparison of Information Regimes

To determine under which conditions the owner benefits from a truthful report on the marginal production costs under the presence of a monopoly supplier, we compare the firm's expected profits for the two information regimes. Installing an information system is profitable, if and only if Πi=Πi(wi=wi)Πn=Πn(wn=wn). We first consider the case where C<Cˆ. Comparing the expected profits Πi in Lemma 4.2, (ii) and Πn in Lemma 4.1 shows that there exists a cutoff value, C_=2a(1φ)/(6(3φ)φ), with 0<C_<Cˆ=a(1φ) such that Πi<Πn for 0<C<C_ and Πi>Πn for C_<C<Cˆ.1313 The observation that C_<Cˆ follows easily from CˆC_=a(1φ)2(4φ+(2φ)φ)/(63φ+φ2)>0. While in the region C_<C<Cˆ the owner benefits from installing an information system, the supplier and the consumers are worse off. Nevertheless, it can be shown that there exists a threshold Cw such that the total supply chain profit and welfare are higher if the firm installs the information system if Cw<C<Cˆ. Details are available upon request.View all notes The following proposition summarizes our findings. Again, the proof is straightforward and is, therefore, omitted.

Proposition 4.3

Consider the range of marginal costs C where 0CCˆ. Then, there exists a value C_=2a(1φ)/(6(3φ)φ) with 0<C_<Cˆ=a(1φ) such that the following results hold.

  1. If C<C_, the firm's expected profit with a monopoly supplier is higher if the owner does not install an information system and relies on ex-ante information about marginal costs, i.e., Πn>Πi.

  2. If C>C_, the firm's expected profit is higher if the owner installs an information system and asks the manager for a cost report, i.e., Πn<Πi.

  3. For C = 0 and C=C_ the owner is indifferent between these two options.

Next, we consider the case where C>Cˆ. A comparison of the expected profits Πi in Lemma 4.2, (iii) and Πn in Lemma 4.1 shows that there exists a cutoff value, C¯=a/(1+φ) which is smaller than Cmax=a(1φ) only if φ<(35)/2. In this case, the firm prefers to install an information system if C>C¯ and prefers not to install an information system if C<C¯. The next proposition summarizes our findings.

Proposition 4.4

Consider the range of costs C where Cˆ<Ca(1φ). Then, the following results hold.

  1. If the probability φ of low marginal cost c = 0 is sufficiently low, i.e., φ<(35)20.38197, then there exists a value C¯=a/(1+φ)<Cmax=a(1φ) such that for C<C¯ the firm's expected profit is higher if the owner does not install an information system and for C>C¯ the firm's expected profit is higher if the owner installs an information system and asks the manager for a cost report. For C=C¯ there is indifference.

  2. If the probability φ is sufficiently high, i.e., φ>(35)2, the firm's expected profit is always higher if the owner does not install an information system.

  3. If the probability φ satisfies φ=(35)/2, the firm's expected profit is always higher if the owner does not install an information system for all Ca(1φ). For C=a(1φ), the firm's expected profit is identical under both systems.

The full range of the owner's choice of information system in the first stage of our game is obtained by combining the results provided in Propositions 4.3 and 4.4. Figure 4 illustrates the situation for φ<(35)/2.

Figure 4. The figure depicts the firm's expected profits Πn (solid curve) and Πi (dash-dotted curve) under endogenous supplier pricing as functions of C for a = 6 and φ=0.3. Between C_ and Cˆ and above C¯, we have Πi>Πn. Hence, it is optimal to install an information system (IS). In the regions between 0 and C_ and between Cˆ and C¯, it is optimal not to install the information system (noIS).

If marginal costs C are small, C<C_, it is optimal not to install an information system since Πn>Πi. If C is medium-low, i.e., C_<CCˆ, it is optimal to install an information system and ask the manager for a cost report. For C=C_ both information regimes yield the same expected profit for the firm. For Cˆ<C<C¯, it is again optimal not to install an information system. Finally, if marginal costs are very high, i.e., C>C¯, it is again optimal to install an information system and ask the manager for a cost report. For C=C¯ both information regimes again yield the same expected profit for the firm.

4.4. Influence of Monopoly Supplier Pricing on the Choice of Information System

There are two major differences that occur due to endogenous supplier pricing.1414 We consider the scenario under a perfect information system separately in the next section.View all notes First, the structure of the optimal choice of the information system changes. While for a fixed input price there is a single threshold such that installing an information system and asking the manager for a cost report is beneficial if marginal costs are sufficiently large, this sort of ‘monotonicity’ is lost with an endogenous supplier price. Second, installing an information system and involving the manager might be beneficial more or less frequently depending on the range of marginal costs.

Consider the change in structure first. As described in Proposition 3.1, with an exogenous input price installing an information system is beneficial if and only if the marginal costs C are sufficiently high, i.e., C>Cb. Intuitively, this is the case if the value of obtaining true information about marginal costs dominates the negative effect of the manager's information rent. With an endogenous supplier price, Propositions 4.3 and 4.4 jointly show that for φ<(35)/2 there are two disconnected regions, C_<C<Cˆ and C¯<C<a(1φ), in which it is optimal to install the information system.

To understand the influence of the monopoly supplier on the design of the information system, we reconsider the composition of expected profits (see Section 3.3) but with endogenous input prices instead of exogenous input costs. The owner benefits from installing an information system if and only if Πi=(x¯n(wi=wi))2+14Var(cˆ)>(xn(wn=wn))2=Πn, where we recall that x¯n denotes the quantity an uninformed firm owner would produce given a cost structure that incorporates the agency costs. In the case of an exogenous input price w, the expressions on both sides of the inequality are evaluated at the same input price while in the case of an endogenous supplier price, the two sides of the inequality have to be evaluated with different input prices, wi and wn. Figure 5 depicts the input prices charged by the supplier depending on the firm's choice of installing an information system (see also Lemmas 4.2, (i) and 4.1). The dashed line represents the input price if the owner does not install an information system and relies on ex-ante cost information, wn. The bold line represents the input price if the owner installs an information system and asks the manager for a cost report, wi. For CCˆ, the input price is lower if the owner installs an information system, while the opposite holds for C>Cˆ.

Figure 5. The figure shows the input price wn if no information system is installed (dashed line) and the input price wi if an information system is installed (bold line) as a function of the costs C for a = 6 and φ=0.4.

Under endogenous supplier pricing, in addition to the value of learning the true marginal costs and the information rent effect discussed earlier, there is a third effect – the input price effect. The input price wi is discontinuous in the marginal costs C, i.e., it jumps upwards at C=Cˆ. Therefore, the input price effect of installing an information system is positive for CCˆ and negative for C>Cˆ. In the region CCˆ, the supplier anticipates that the owner faces agency costs if an information system is installed. In effect, this leads to higher marginal production costs and to a decrease in the expected production quantity, φxi(0)+(1φ)xi(C)=(aC)/4<(a(1φ)C)/4=xn. Therefore, the supplier reduces its input price to increase demand for the input. The result is that C_<Cb.1515 Assume the opposite, i.e C_>Cb. Rewrite the inequality Πi(wi=wi)<Πn(wn=wn) as Πi(wi=wi)Πi(wn=wn)+Πi(wn=wn)<Πn(wn=wn) and evaluate at C=Cb. Since Πi(wn=wn)=Πn(wn=wn), it follows that Πi(wi=wi)Πi(wn=wn)<0, which is a contradiction because of wi<wn.View all notes. For C>Cˆ, the supplier accepts that the firm does not produce in case of high costs if the information system is installed. Therefore, it charges the monopoly input price given that c = 0. This input price is higher than if the owner does not install an information system. Now the owner's value of learning the true marginal costs has to be sufficiently large to dominate this negative input price effect and the agent's information rent which is the case if C>C¯ and C¯>Cˆ. To summarize, the input price effect causes a structural change in the owner's decision to install an information system.

Consider next the question whether the firm would want to install an information system more or less frequently in the presence of a monopoly supplier. The arguments above show that for CCˆ, we have C_<Cb=2(1φ)(aw)/((φ3)φ+3). Consequently, we can predict that installing an information system is beneficial more frequently. Consider next the case C>Cˆ. Obviously, Cb=(aw)/(1+φ)<C¯=a/(1+φ). Therefore, we can predict that installing the information system is beneficial less frequently.

4.5. Firm Installs Perfect Information System

In this section, we want to address another main point of our analysis: would the firm owner prefer installing a perfect information system and learn the true cost information at cost K without involving the manager? Recall that in the case of an exogenous supplier price the owner would certainly prefer such a system if K = 0. Here we want to demonstrate that an endogenous supplier price changes this result. Notice that this research question is somehow opposite to our analysis so far as well as Rajan and Saouma (2006 Rajan M. V., & Saouma R. E. (2006). Optimal information asymmetry. The Accounting Review, 81(3), 677712.[Crossref], [Web of Science ®] [Google Scholar]). In Rajan and Saouma (2006 Rajan M. V., & Saouma R. E. (2006). Optimal information asymmetry. The Accounting Review, 81(3), 677712.[Crossref], [Web of Science ®] [Google Scholar]), the question is how much the principal would want to inform the manager given that the principal does not have information while here we ask whether the principal prefers to be informed taking into account that if the principal does not obtain the information, the manager could be asked for a cost report instead. The following result can be shown.

Lemma 4.5

If the firm owner installs a perfect information system at costs K, then there exists a value Cˆ=a/(1+φ) such that the following holds.

  1. For CCˆ the firm's optimal quantities are given by xI(0)=(a+(1φ)C)/4 and xI(C)=(a(1+φ)C)/4. The supplier charges the input price wI=(a(1φ)C)/2. The firm's equilibrium profit is ΠI=φ(xI(0))2+(1φ)(xI(C))2K and the supplier's equilibrium profit is ΠIs=wI(φxI(0)+(1φ)xI(C)).

  2. For C>Cˆ, the firm's optimal quantities are given by xI(0)=a/4 and xI(C)=0. The supplier charges the input price wI=a/2. The firm's equilibrium profit is ΠI=φ(xI(0))2K and the supplier's equilibrium profit is ΠIs=wI(φxI(0)).

Note that CˆCˆ=a(1φ) with strict inequality for φ0. In fact, Cˆ<a(1φ) if and only if φ<12(35)0.38197. So the second case of Lemma 4.5 only occurs for φ<12(35).

Using Lemma 4.5, we can now demonstrate that although the firm owner has to pay an information rent if only the manager can observe the firm's marginal costs, in the presence of a third party (here a supplier) the owner might nevertheless prefer this situation rather than installing a perfect information system which grants the owner access to cost information. Intuitively, the firm's “handicap” works to the owner's advantage since the commitment to rely on the manager's cost report influences the supplier's price response.

To elaborate, consider the range CCˆ and compare the firm's expected profit if a perfect information system is installed, ΠI, with the expected profit if the owner relies on the manager's cost report, Πi; see Lemmas 4.5 and (4.2), respectively. We get ΠIΠi=φC2a(1φ)C2+5φ3φ216(1φ)K. Assume that K = 0 as in the case with exogenous supplier price. The expression on the right-hand side is concave in C and is zero at C = 0 and C~=2a(1φ)/(2φ)(1+3φ). Since C_<C~ and C~<Cˆ if φ>(733)/60.2092, if the probability φ of marginal costs c = 0 is sufficiently high there is a range of marginal costs, C~<C<Cˆ, where the firm's expected profit is higher if it relies on the manager's cost report. The surprising conclusion which contrasts conventional wisdom is that even if the owner could learn the true marginal costs by installing a perfect information system at no costs, it might not be beneficial.

Proposition 4.6

If the probability φ is sufficiently large, i.e., φ>(733)/6, then there exists a value C~=2a(1φ)/(2φ)(1+3φ)>C_ such that for C~<CCˆ the firm prefers to install an information system and ask the manager for a cost report even if perfect information would be available at no costs (i.e., Πi>ΠI for K=0). For C = 0 the owner is indifferent and for 0<C<C~, the owner prefers a perfect information system.

To end this section, we briefly comment on the case of positive information costs K and still assume φ>(733)/6 for simplicity. The difference ΠIΠi is concave in C and has a unique maximum. Obviously, if K is larger than the maximum, it is never optimal to install a perfect information system. However, if K is sufficiently small, there exist values C~1(K) and C~2(K) such that it is optimal to install an information system and ask the manager for a cost report if and only if C~1(K)<C<C~2(K). If C<C~1(K) or C>C~2(K), it is optimal to install a perfect information system.1616 Note that the difference is negative at C = 0 and also at C=Cˆ (because of φ>(733)/6). If φ>(733)/6 and K sufficiently small, installing a perfect information system is optimal for C(C~1(K),Cˆ). For larger values of K, installing a perfect information system is optimal if and only if C~1(K)<C<C~2(K).View all notes Finally, due to the concavity of the difference ΠIΠi it follows that C~1(K) is increasing in K, while C~2(K) is decreasing in K.

5. Robustness and Extensions

Our results have been derived in a parsimonious setting with a binary cost structure assuming c = 0 and c = C>0. This raises the question how robust the main insights of this paper are with regard to modifications of the cost structure. We first report our findings if both marginal cost realizations are positive. Then we briefly summarize the results for the case of three possible cost realizations. Finally, we illustrate how our results are affected if cost realizations are continuously distributed over an entire interval.1717 To save space, we will not provide the full details of the derivations. They are, however, available upon request.View all notes

5.0.1. Positive binary cost realizations

Assume that marginal costs can be 0<c1<c2 with probabilities φ and 1φ. Then it can be shown that wn=(aφc1(1φ)c2)/2. Fixing c1, we get wi=(ac2)/2 if c2<cˆ2 and wi=(ac1)/2 if c2>cˆ2 where the threshold cˆ2=a(ac1)φ. It can now be shown that there exist c_2 and c¯2 such that identical statements as in Propositions 4.3 and 4.4 can be made. Moreover, if the probability of low costs is sufficiently large, the owner is better off installing an information system and asking the manager for a cost report rather than installing a perfect information system.

5.0.2. Three possible cost realizations

Assume that marginal costs can be c1=0, c2=C/2, and c3=C with probabilities φ1, φ2 and 1φ1φ2 respectively, so that the expected marginal costs are E(c)=C(1φ1φ2/2). Then wn=(aE(c))/2. The case where the owner installs an information system and asks the manager for a cost report can be analyzed using standard textbook methods (e.g., Laffont & Martimort, 2002 Laffont J.-J., & Martimort D. (2002). The theory of incentives. the principal-Agent model. Princeton, NJ: Princeton University Press.[Crossref] [Google Scholar]).1818 To guarantee that the monotonicity condition x(0)<x(C/2)<x(C) holds, we need φ2φ1(1φ1φ2).View all notes We find that there are three intervals for C.1919 For example, if φ1=φ2=1/3, then the thresholds where the input price changes are Cˆ10.31a and Cˆ20.5857a.View all notes For low C, the firm produces no matter what the cost realization is and the supplier charges wi=(aC)/2. If C is intermediate, then the firm shuts down production only if costs are C and the input price is wi=(2aC)/4. Finally, if marginal costs are high, then the firm only produces if marginal costs are low and the input price is wi=a/2. In each of the intervals, the owner prefers not to install an information system if C is small, but prefers to install an information system and asks the manager for a report if C is sufficiently large.2020 In the case of equal probabilities, the thresholds where the transition occurs are C0.293a, C0.522a and C0.845a. Hence, for example in the case where all quantities are positive, if C<0.293a the owner prefers not to install the information system while it is profitable to install an information system for 0.293a<C<0.31a.View all notes Finally, if we restrict ourselves to the case where all quantities are positive, then in line with our simple case considered in the main text, we can find situations where the owner might prefer to install an information system and ask the manager for a cost report rather than install a perfect information system. The conclusion of our analysis is that the main messages of our paper carry over to the case of three possible cost realizations.

5.0.3. Continuous distribution of marginal costs

We assume that marginal costs c are uniformly distributed on the interval [0,C]. If the firm owner does not install an information system, the production decision can only be conditioned on the expected marginal costs, E(c)=C/2. The optimal quantity is, hence, xn=14(2aC2w). The supplier maximizes Πis=wxn, which yields wn=14(2aC). The firm's resulting expected profit is Πn=116(aC2)2.

If the firm owner installs an information system, then it is again standard to show that if the manager obtains the cost information, the owner's problem can be simplified to2121 In the case where the owner installs a perfect information system, the optimization problem is maxx()0C(ax(c)cw)x(c)(1/C)dc. The difference to the present situation where the owner installs an information system and asks the manager for a cost report, is that here we have costs of 2c instead of c.View all notes maxx()0C(ax(c)2cw)x(c)1Cdc. Pointwise optimization yields xi(c)=12(a2cw) as long as xi(c)>0 (i.e., for c<(aw)/2) and 0 otherwise. Anticipating the firm's demand x(c), the supplier's objective function is 1C0(aw)/212w(a2cw)dc=w(aw)28C for a>w>a2C and 1C0C12w(a2cw)dc=12w(aCw)forwa2C, and 0 for w>a. Using the corresponding first-order conditions, it can be shown that wi=(aC)/2 for Ca3 and wi=a3 for C>a3. In contrast to the case of a discrete probability distribution, the input price wi(C) depends continuously on C.

The owner's objective function can be written as 0C(x(c))2(1/C)dc and inserting the corresponding input price wi, the firm's expected profit is Πi=148(3a26aC+7C2) for C<a/3 and Πi=(a3/81C) for Ca/3. Since the input price wi() is continuous, the firm's expected profit Πi() is continuous in C as well.

It is easy to see that for C<a/3, the firm's corresponding profit Πi is always strictly smaller than Πn (except for C = 0 where they are equal). For Ca/3, it can be shown that there is a unique intersection C¯ of Πn and Πi between a and 2a.2222 To see this, note that the profit difference ΠnΠi is a polynomial of order 3 and is positive for C=a/3 while it goes to for C0. Hence, there is one root in (0,a/3). The difference is negative for C = 2a and positive for C = 3a. Hence, one further root is in (2a,3a). Since the difference is positive for C = a, the final root is in (a,2a).View all notes

The following proposition summarizes the result.

Proposition 5.1

There exists a value C¯(a,2a) such that for C<C¯ the firm's expected profit is higher if the firm owner does not install an information system and for C>C¯ the firm's expected profit is higher if the firm owner installs an information system and asks the manager for a cost report.

This result differs qualitatively from the more complex result in the discrete case (cf. Proposition 4.4). To identify the reason for this difference, let φ=12 in Proposition 4.4 which corresponds to a discrete uniform distribution. First, notice that in both cases of cost distributions (discrete or continuous), the firm's expected profit if it does not install an information system is Πn=116(aC/2)2. Only expected marginal costs matter here, and they are identical in both models (given φ=12). If the firm installs an information system, examining the respective optimization problem shows that the expected costs (taking into account the agency problem) are the same (and equal to C) in both models. For both cases, we find two regimes (the role of Cˆ where the supplier changes the input price due to rationing is played by C=a/3 here) where in the first regime the firm always produces in equilibrium independent of marginal costs while in the second regime the firm stops producing for high values of c. Also notice that for C<a3, in both cases the input price is wi=aC2 and, hence, is smaller than the input price wn=14(2aC). The input price effect of installing an information system is positive. If Ca/3, then the input price is larger than wn if C>2a/3. Consequently, if C is sufficiently large, the input price effect of installing an information system is negative just like in the discrete distribution model.

Despite these parallels, in the case of a continuous cost distribution it is not profitable for the owner to install an information system for C<a/3 and only if C is sufficiently large is it profitable to do so (akin to the discrete distribution model for C>Cˆ). The difference in the results is caused by a smaller expected value of perfect information in the continuous model. In the continuous model, the variance is 0C(cC)2(1/C)dc=C23 while in the discrete model the variance equals 12(0C)2+12(2CC)2=C2. Since the expected value of perfect information increases with the variance, the claim follows. Intuitively, the distance to the expected value is always maximal in the discrete case (C), while it can be rather small in the continuous case. Consequently, for C<a/3 the function Πi is (point-wise) larger in the discrete case than in the continuous case.

To conclude our analysis, we would like to comment on the relation between the continuous model and the model with a discrete distribution of marginal costs. Although we believe that it is theoretically paying to study the continuous model, we also think that the discrete model is more in line with the real world. Note that in the continuous model it is assumed that the manager obtains perfect cost information for any possible cost realization in the considered interval. One possible way to model an information environment which is less demanding and requires less precise information, is via a partition (for example, the interval [0,C] could be partitioned into the two intervals [0,z] and (z,C] with 0<z<C). The information system then tells the agent only the interval in which the true costs lie. Such a partition transfers the continuous model into a discrete model. Since information systems are in general not completely precise (and often take the form of partitions), it seems plausible that in most cases information will be discrete.2323 See, for example, Bertomeu, Magee, and Schneider (2019 Bertomeu J., Magee R., & Schneider G. (2019). Voting over disclosure standards. European Accounting Review, 28(1), 4570.[Taylor & Francis Online], [Web of Science ®] [Google Scholar]) where accounting standards are modeled as partitions. The underlying information, however, is continuous in their model.View all notes

6. Conclusion

Management accounting research has mainly focused on questions concerning the design of control and incentives within firm boundaries. Broadly speaking, our paper makes the point that a firm interacts with a variety of external non-financial stakeholders like customers, suppliers, and rivals. The decisions that are made within the firm have an impact on the behavior of other stakeholders and, vice versa, decisions made by these external stakeholders influence decisions within the organization. In this sense, we contribute to the emerging line of – mainly empirical – accounting research that studies the interaction between a firm's corporate policy and the firm's non-financial stakeholders (e.g., Arora & Alam, 2005 Arora A., & Alam P. (2005). CEO compensation and stakeholders claims. Contemporary Accounting Research, 22(3), 519547.[Crossref], [Web of Science ®] [Google Scholar]; Dekker, 2016 Dekker H. C. (2016). On the boundaries between intrafirm and interfirm management accounting research. Management Accounting Research, 31, 8699.[Crossref], [Web of Science ®] [Google Scholar]; Hui, Klasa, & Yeung, 2012 Hui K. W., Klasa S., & Yeung P. E. (2012). Corporate suppliers and customers and accounting conservatism. Journal of Accounting and Economics, 53(1/2), 115135.[Crossref], [Web of Science ®] [Google Scholar]; Raman & Sharur, 2008 Raman K., & Sharur H. (2008). Relationship-specific investments and earnings management: E evidence on corporate suppliers and customers. The Accounting Review, 83(4), 10411081.[Crossref], [Web of Science ®] [Google Scholar]; Sedatole, Vrettos, & Widener, 2012 Sedatole K., Vrettos D., & Widener S. (2012). The use of management control mechanisms to mitigate moral hazard in the decision to outsource. Journal of Accounting Research, 50(2), 553592.[Crossref], [Web of Science ®] [Google Scholar]).

The common view in the accounting and economics literature is that the owner benefits from involving a better-informed manager in decisions but that the manager has an incentive to misreport private information. Eliciting truthful information, the owner has to pay an information rent which makes the manager's participation costly. Depending on the owner's value of perfect information information and the size of the manager's information rent, the owner prefers to rely on ex-ante information or to install an information system and let the manager report the observed costs. This paper offers a novel perspective on the optimal design of a firm's information system since it highlights the important interaction between the choice of information system and the firm's supply side. We show that there is a crucial interdependence between the design of the firm's information system and the pricing behavior of the firm's supplier. In particular, if the owner asks the manager for a cost report, the associated information rent reduces the firm's expected production. This causes the supplier to reduce the input price. This positive input price effect has to be traded off against the other two, more well-known effects.

Acknowledgments

Our paper has benefited from detailed and insightful comments of Robert Göx (editor) and two anonymous reviewers. For helpful comments on earlier versions of the paper, we would like to thank the seminar participants at the University of Mannheim and the University of Barcelona, participants of the EAA annual conferences in Valencia 2017 and Paphos 2019, of the EARIE Conference 2019 in Barcelona, the ACMAR Conference 2019, the Annual VHB Conference 2019 in Rostock and the ARFA Workshop 2019 in Basel. We also thank Ralf Ewert, Clemens Löffler, Stefan Reichelstein, Ulrich Schäfer, Ulf Schiller, Dirk Simons, Alfred Wagenhofer and Katrin Weiskirchner-Merten for detailed remarks and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix. Proofs

Proof

Proof of Lemma 4.2

To prove the lemma, we make use of the following lemma:

Lemma A.1

The locally optimal input price of the supplier is given by (i) if 0<waC/(1φ), we have w=(aC)/2 for Ca(1φ)/(1+φ) and w=aC/(1φ) for C>a(1φ)/(1+φ); (ii) if aC(1φ)<wa, we have w=a/2 for C>a(1φ)/2 and w=aC/(1φ) for Ca(1φ)/2.

Proof

Proof of Lemma A.1

We first derive the supplier's locally optimal input price in region 1 (i.e.,waC/(1φ)). The objective function is given by the first line in (3), which can be rewritten as 12(aCw)w. The first-order condition yields the input price w=(aC)/2 as long as this value is in region 1, i.e., as long as w=(aC)/2aC/(1φ). This latter condition translates into the requirement that Ca(1φ)/(1+φ). Since the objective function in region 1 is concave, this is the locally optimal value in region 1 as long as w is in region 1. If this is not the case, then the locally optimal input price in region 1 is attained at the upper bound of the region (w=aC/φ). The result for region 2 can be obtained similarly.

We can now prove Lemma 4.2. First, note that a(1φ)/(1+φ)>a(1φ)/2. Therefore, for Ca(1φ)/2 the local maximum in region 1 is the global maximum. This is because the objective function decreases in w if w exceeds (aC)/2 in region 1 and because the objective function is continuous in w and must decrease in region 2 (because the local maximum in region 2 is attained at the lower bound of region 2). Similarly, one can show that for Ca(1φ)/(1+φ) the local maximum in region 2 is the global maximum. Finally, for a(1φ)/(1+φ)>C>a(1φ)/2 the maximal value in region 1 is 18(aC)2 and in region 2 it is φ(a2/8). For C=a(1φ)/2, we have 18(aC)2>φ(a2/8) and for C=a(1φ)/(1+φ) we have 18(aC)2<φ(a2/8). Also, 18(aC)2 is strictly decreasing in C while φ(a2/8) is constant in C. Therefore, there exists a unique value Cˆ such that 18(aCˆ)2=φ(a2/8). Solving this equation yields Cˆ=a(1φ).

Proof

Proof of Proposition 4.4

It remains to be shown that C¯=a/(1+φ)<Cmax=a(1φ) if and only if φ<(35)/2. It is not difficult to show that 1/(1+φ)=1φ if and only if φ=0 or φ=(35)2. Substituting φ=Q2 yields 1(1φ)(1+φ)=Q(Q2+Q1). This polynomial is strictly positive for Q = 1 and its derivative QQ(Q2+Q1)=1+Q+Q2+Q(1+2Q) is negative for Q = 0. Therefore 11+φ<1φ if and only of 0<φ<352. This finishes the proof.

Proof

Proof of Lemma 4.5

The proof follows along the lines of the proof of Lemma 4.2. The value Cˆ is the unique solution of the equation 18(aC(1φ))2=φ(a2/8) that satisfies aC(1φ)>0. We have Cˆ=a(1φ)<Cˆ=a1+φ if and only if a(1φ)<a which is satisfied for all φ0.

Proof

Proof of Proposition 4.6

For CCˆ we can calculate ΠIΠi=Cφ(2a(φ1)+C(3φ2+5φ+2))/16(φ1). This difference is strictly negative if and only if C>C~:=2a(1φ)/(2φ)(1+3φ). It remains to compare the threshold value C~ with C_ and Cˆ. Recall that C_=2a(1φ)/(6(3φ)φ) and Cˆ=a(1φ). Comparing the first two values, we get C~C_=8(1φ)3/(2φ)(1+3φ)(6(3φ)φ)0. For the second comparison, we have C~Cˆ=(2a(1φ)/(2φ)(1+3φ))a(1φ). This difference equals 0 if and only if 2(φ1)(1φ)(3φ25φ2)=0. It is not difficult to show that the above polynomial has exactly three zeros, φ=1,φ=0, and φ=16(733), and that the derivative of C~Cˆ is ∞ for φ=0. Also, substituting φ=Q2 in this polynomial yields (as is easily checked) the expression (Q1)2Q(3Q2+3Q2). Since this expression has a single root at 16(333), the derivative cannot be zero at 16(333). Therefore, it must be strictly increasing or strictly decreasing at Q=16(333) and (since a polynomial is continuously differentiable) this also holds in a neighborhood of Q=16(333). Since it is strictly positive for Q<16(333) it must be decreasing and therefore for Q>16(333) the polynomial is negative. This translates directly to the difference C~Cˆ. Therefore, C~Cˆ is negative for φ>7336.

Notes

1 In our setting, while the principal controls the information flow, information is exogenous. For work on endogenous information structure where the agent can expend costly effort to acquire information, see, e.g., Kessler (1998 Kessler A. S. (1998). The value of ignorance. The RAND Journal of Economics, 29(2), 339354.[Crossref], [Web of Science ®] [Google Scholar]), Hoppe (2013 Hoppe E. I. (2013). Observability of information acquisition in agency models. Economics Letters, 119(3), 104107.[Crossref] [Google Scholar]), Cremer, Khalil, and Rochet (1998 Cremer J., Khalil F., & Rochet J.-C. (1998). Contracts and productive information gathering. Games and Economic Behavior, 25, 174193.[Crossref], [Web of Science ®] [Google Scholar]). However, these papers do not consider the influence of an external party on optimal information provision.

2 See Taylor and Xiao (2010 Taylor T. A., & Xiao W. (2010). Does a manufacturer benefit from selling to a better-forecasting retailer. Management Science, 56(9), 15841598.[Crossref], [Web of Science ®] [Google Scholar]) for a similar result in a manufacturer–retailer setting.

3 Likewise Cremer (1995 Cremer J. (1995). Arm's length relationships. The Quarterly Journal of Economics, 110(2), 275295.[Crossref], [Web of Science ®] [Google Scholar]2010 Cremer J. (2010). Arm's length relationships without moral hazard. Journal of the European Economic Association, Proceedings of the Twenty-Fourth Annual Congress of the European Economic Association, 8(23), 377387. [Google Scholar]) demonstrates that in agency settings lowering the costs of information acquisition can actually worsen the situation of the principal.

4 Our results qualitatively hold for realizations c1>0 and c2>0. Furthermore, they also hold for three possible cost levels, e.g., c1=0,c2=C/2,c3=C with positive probabilities φ1,φ2,1φ1φ2. In Section 5, we provide more details and further describe the changes for the case where realizations of marginal costs are continuously distributed over an interval [0,C].

5 This is in line with the general idea of slack in the literature captured by paying an information rent to the manager to induce truthful reporting of the manager's private information (e.g., Schiff & Lewin, 1970 Schiff M., & Lewin A. Y. (1970). Impact of people on budgets. The Accounting Review, 45(2), 259268.[Web of Science ®] [Google Scholar]).

6 The subscript n denotes the information scenario where no player has information about the true marginal costs.

7 See, for example, Laffont and Martimort (2002 Laffont J.-J., & Martimort D. (2002). The theory of incentives. the principal-Agent model. Princeton, NJ: Princeton University Press.[Crossref] [Google Scholar]).

8 The subscript i denotes the information scenario where the owner obtains information about the true marginal costs from the manager.

9 The subscript I denotes the information scenario where the owner obtains information about the true marginal costs by installing a perfect information system.

10 In order to guarantee that Cb<Cmax=a(1φ), the condition (1φ)(1+φ)1>w/a has to be fulfilled.

11 In the case w>a, the supplier's expected profit is zero. Such a high input price is never optimal, however, since the supplier could achieve a positive profit by setting an input price w<a. Therefore, we can restrict our analysis to the interval 0<w<a.

12 The firm's quantities follow from inserting the input price wi into xi(0)=12(aw) and xi(C)=12(awC/(1φ)), where xi(C)=0 for C>Cˆ. Using the input price and the firm's quantities, the expected profits can be derived.

13 The observation that C_<Cˆ follows easily from CˆC_=a(1φ)2(4φ+(2φ)φ)/(63φ+φ2)>0. While in the region C_<C<Cˆ the owner benefits from installing an information system, the supplier and the consumers are worse off. Nevertheless, it can be shown that there exists a threshold Cw such that the total supply chain profit and welfare are higher if the firm installs the information system if Cw<C<Cˆ. Details are available upon request.

14 We consider the scenario under a perfect information system separately in the next section.

15 Assume the opposite, i.e C_>Cb. Rewrite the inequality Πi(wi=wi)<Πn(wn=wn) as Πi(wi=wi)Πi(wn=wn)+Πi(wn=wn)<Πn(wn=wn) and evaluate at C=Cb. Since Πi(wn=wn)=Πn(wn=wn), it follows that Πi(wi=wi)Πi(wn=wn)<0, which is a contradiction because of wi<wn.

16 Note that the difference is negative at C = 0 and also at C=Cˆ (because of φ>(733)/6). If φ>(733)/6 and K sufficiently small, installing a perfect information system is optimal for C(C~1(K),Cˆ). For larger values of K, installing a perfect information system is optimal if and only if C~1(K)<C<C~2(K).

17 To save space, we will not provide the full details of the derivations. They are, however, available upon request.

18 To guarantee that the monotonicity condition x(0)<x(C/2)<x(C) holds, we need φ2φ1(1φ1φ2).

19 For example, if φ1=φ2=1/3, then the thresholds where the input price changes are Cˆ10.31a and Cˆ20.5857a.

20 In the case of equal probabilities, the thresholds where the transition occurs are C0.293a, C0.522a and C0.845a. Hence, for example in the case where all quantities are positive, if C<0.293a the owner prefers not to install the information system while it is profitable to install an information system for 0.293a<C<0.31a.

21 In the case where the owner installs a perfect information system, the optimization problem is maxx()0C(ax(c)cw)x(c)(1/C)dc. The difference to the present situation where the owner installs an information system and asks the manager for a cost report, is that here we have costs of 2c instead of c.

22 To see this, note that the profit difference ΠnΠi is a polynomial of order 3 and is positive for C=a/3 while it goes to for C0. Hence, there is one root in (0,a/3). The difference is negative for C = 2a and positive for C = 3a. Hence, one further root is in (2a,3a). Since the difference is positive for C = a, the final root is in (a,2a).

23 See, for example, Bertomeu, Magee, and Schneider (2019 Bertomeu J., Magee R., & Schneider G. (2019). Voting over disclosure standards. European Accounting Review, 28(1), 4570.[Taylor & Francis Online], [Web of Science ®] [Google Scholar]) where accounting standards are modeled as partitions. The underlying information, however, is continuous in their model.

 

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