Decentralized capacity management and internal pricing

3.1 Benchmark scenario: dedicated capacity

If the production processes of the two divisions are sufficiently different, it may be impossible for the divisions to redeploy the available capacity stock k _t in period t after the random shocks \(\epsilon\) _t are realized. We refer to such a setting as one of dedicated capacity as Division i’s initial capacity assignment k _it , made at the beginning of period t, is also equal to the capacity ultimately available for its use in that period. Put differently, capacity assignments can only be altered at the beginning of each period but not within a period.

Lemma 1 shows that in the dedicated capacity scenario the firm’s optimization problem is separable not only cross-sectionally across the two divisions but also intertemporally.¹¹ The non-negativity constraints for new investments, b _t ≥ 0, will not bind provided the corresponding sequence of capacity levels k = (k ₁, k ₂, ...) satisfy the monotonicity requirement k _t+1 ≥ k _t for all t. This latter condition will be met whenever the expected marginal revenues satisfy the monotonicity condition in (2).¹²

Suppose now the firm depreciates investments according to the relative practical capacity rule and the transfer price for capacity services charged to Division 2 is based on the full historical cost (which includes the imputed interest charges).¹⁵ As a consequence, both divisions will be charged the competitive rental price c per unit of capacity in each period. The key difference in the treatment of the two divisions is that the downstream division can rent capacity on an “as needed” basis, while capacity investments entail a multi-period commitment for the upstream division. In making its capacity investment decision in the current period, the upstream division has to take into account the resulting historical cost charges that will be charged against its performance measures in future periods. Given the weights u _t that the divisions attach to their periodic performance measures, we then obtain a multi-stage game in which each division makes one move in each period; that is, each division chooses its capacity level.

As demonstrated in the proof of Proposition 1, the divisional managers face a T-period game with a unique subgame perfect equilibrium.¹⁶ Irrespective of past decisions, the downstream division has a dominant strategy incentive to secure the optimal capacity level because it is charged the relevant unit cost c. The upstream division potentially faces the constraint that, in any given period, it may inherit more capacity from past investment decisions than it currently needs. However, provided the divisions’ marginal revenues are increasing over time; that is, condition (2) is met, the upstream division will not find itself in a position of excess capacity, provided the downstream division follows its dominant strategy.¹⁷

The result in Proposition 1 makes a strong case for full-cost transfer pricing, that is, a transfer price that comprises variable production costs (effectively set to zero in our model) plus the allocated historical cost of capacity, c. Survey evidence indicates that in practice full cost is the most prevalent approach to setting internal prices. In our model, the full cost rule leaves the upstream division with zero economic (residual) profit on internal transactions and, at the same time, provides a goal congruent valuation for the downstream division in its demand for capacity.

As argued by Balakrishnan and Sivaramakrishnan (2002), Goex (2002), Bouwens and Steens (2008) and others, it has been difficult for the academic accounting literature to justify the use of full-cost transfers. Most existing models have focused on one-period settings in which capacity costs were taken as fixed and exogenous. As a consequence, full-cost mechanisms typically run into the problem of double marginalization; that is, the buying entity internalizes a unit cost that exceeds the marginal cost to the firm. Some authors, including Zimmerman (1979), have suggested that fixed cost charges are effective proxies for opportunity costs arising from capacity constraints. This argument can be made in a “clockwork environment” in which there are no random disturbances (i.e., \(\epsilon_t\equiv \bar{\epsilon}_t\)). At date t, the cost of capacity investments for that period is sunk, yet the opportunity cost of capacity is equal to c, precisely because at date t − 1 each division secured capacity up to the point where its marginal revenue is equal to c. Once there are random fluctuations in the divisional revenues, however, there is no reason to believe that the opportunity costs at date t relate systematically to the historical fixed costs at the earlier date t − 1.

Our rationale for the use of full cost transfer prices hinges crucially on the dynamic of overlapping capacity investments. Since the firm expects to operate at capacity, divisional managers should internalize the incremental cost of capacity; i.e., the unit cost c. The relative practical capacity depreciation rule ensures that the unit cost of both incumbent and new capacity is valued at c in each period. As a consequence, the historical fixed cost charges can be “unitized” without running into a double marginalization problem with regard to the acquisition of new capacity.¹⁸

3.2 Fungible capacity

We note that with dedicated capacity the optimal \(\bar{k}^o_{it}\) for each division depends only on θ _it . With fungible capacity, in contrast, the optimal aggregate \(k_t^o\) depends on both θ_1t and θ_2t . The proof of Lemma 2 shows that, for any given capacity level k, the expected shadow prices are increasing over time. As a consequence, the first-best capacity levels given by (12) are also increasing over time, which in turn implies that the non-negativity constraints b _t ≥ 0 again do not bind.

These payoffs ignore the transfer payment c ·k _2t that Division 2 makes at the beginning of the period, since these payoffs are viewed as sunk at the negotiation stage. The total transfer payment made by Division 2 in return for the ex post efficient quantity \( q_2^*(k_t, \theta_t,\epsilon_t)\) is then given c · k _2t + ΔTP. Clearly, ΔTP > 0 if and only if \( q_2^*(k_t, \theta_t,\epsilon_t)> k_{2t}\). We refer to the resulting “hybrid” transfer pricing mechanism as adjustable full cost transfer pricing.

At first glance, the possibility of reallocating the initial capacity rights appears to be an effective mechanism for capturing the trading gains that arise from random fluctuations in the divisional revenues. However, the following result shows that the prospect of such negotiations compromises the divisions’ long-term incentives.

The proof of Proposition 2 shows that, for some performance measure weights u _t , there is no equilibrium that results in efficient capacity investments. In particular, the proof identifies a dynamic holdup problem that results when the downstream division drives up its capacity demand opportunistically in an early period in order to acquire some of the resulting excess capacity in later periods through negotiation. Doing so is generally cheaper for the downstream division than securing capacity upfront at the transfer price c. Such a strategy will be particularly profitable for the downstream division if the performance measure weights u _2t are such that the downstream division assigns more weight to the later periods.¹⁹

It should be noted that the dynamic holdup problem can emerge only if the downstream division anticipates negotiation over actual capacity usage in subsequent periods. In the dedicated capacity scenario examined above, the downstream division could not possibly gain by driving up capacity strategically because it cannot appropriate any excess capacity through negotiation. The essence of the dynamic holdup problem is that the downstream division has the power to force long-term asset commitments without being accountable in the long-term. That power becomes detrimental if the downstream division anticipates future negotiations over actual capacity usage.

6.1 Nonlinear shadow prices

Our first extension concerns the effect of relaxing the linearity assumption in A1. One important implication of A1 is that the optimal aggregate capacity level is the same \(k_t^o\) in both the dedicated and the fungible capacity scenario. The following result shows that the curvature of the shadow price in \(\epsilon\) is crucial in shaping the comparison between \(k_t^o\) and \(\bar k_t^o\).

According to Proposition 5, the curvature of the shadow price determines whether a risk-neutral central decision maker would effectively be risk-seeking or risk-averse with respect to the residual uncertainty associated with the stochastic shock \(\epsilon\) _t . Relative to the benchmark setting of dedicated capacity, in which capacity reallocations are (by definition) impossible, a shadow price function, S(·), that is convex in \(\epsilon\) _t makes the volatility inherent in \(\epsilon\) _t more valuable to a risk-neutral decision maker. The central decision maker would therefore be willing to invest more in capacity. The reverse holds when the shadow price is concave. We point out in passing that the assumption that \(R_i(q_{it}, \theta_{it}, \epsilon_{it}) = \epsilon_{it} \cdot \hat R_i(q_{it}, \theta_{it})\) does not imply the linearity of S(·) in \(\epsilon\) _t , because \(\epsilon\) _t enters S(·) not only directly but also via the ex post efficient capacity allocation, \(q_i^*(k_t, \theta_t,\epsilon_t)\).

The curvature of the shadow price functions hinges (unfortunately) on the third derivatives of the net-revenue functions. All three scenarios identified in Proposition 5 can arise for standard functional forms. For instance, it is readily checked that, if \(R_i(q,\theta_{it}, \epsilon_{it}) = \epsilon_{it} \cdot \theta_{it} \cdot \sqrt{q}\) , then the shadow price is a convex function of \(\epsilon\) _t and therefore \(k_t^0 > \bar k_t^0\). On the other hand, S(·, ·, \(\epsilon\) _t ) is concave when \(R_i(q,\theta_{it}, \epsilon_{it}) = \epsilon_{it} \cdot \theta_{it} \cdot (1- e^{-q})\). As mentioned above, examples of revenue functions that yield linear shadow prices, and hence \(k_t^o = \bar k_t^o\) for each t, include: (i) \(R_i (q, \theta_{it}, \epsilon_{it}) = \epsilon_{it} {\cdot}\theta_{it} {\cdot}{\ln}q \) and (ii) \(R_i (q, \theta_{it}, \epsilon_{it}) = q \cdot \left[\epsilon_{it} \cdot \theta_{it} - h_i \cdot q\right]\). It should be noted that all of the above examples satisfy assumption that the noise term \(\epsilon\) _it enters R(·) multiplicatively and yet S(·) is generally not a linear function of \(\epsilon\) _t .

Returning to an organizational structure of centralized capacity ownership, Propositions 3 and 5 strongly suggest that, if the shadow price function S(·) is not linear in \(\epsilon\) _t , adjustable full cost transfer pricing will no longer result in efficient capacity investments because of a coordination failure in the divisional capacity requests. The following result characterizes the directional bias of the resulting capacity levels.

Transfers at cost lead each division to properly internalize the incremental cost of capacity. However, as noted above, the divisional investment incentives are essentially a convex combination of two forces: the benefits of capacity that a division receives on its own and the optimized revenue that the two divisions can attain jointly by reallocating capacity. When \(k_t^o>\bar k_t^o\) , because the shadow price is convex in \(\epsilon\) _t , the efficient capacity level \(k_t^o\) cannot emerge in equilibrium.²⁸

Transfer pricing surveys indicate that cost-plus transfer prices are widely used in practice. Some authors have suggested that this policy reflects fairness considerations in the sense that both profit centers should view a transaction as profitable (Eccles 1985 and Eccles and White 1988).²⁹ In contrast, our result here points to mark-ups as an essential tool for correcting the bias resulting from the fact that neither division fully internalizes the externality associated with uncertain returns from capacity investments. From the perspective of the firm’s central office, a major obstacle, of course, is that the optimal mark-up depends on the information variables, θ _t , which reside with the divisional managers.

6.2 Appointing a gatekeeper

One conclusion of Proposition 6 above is that even if the firm centralizes capacity ownership and charges the divisions the competitive rental prices of capacity, there will be a remaining coordination problem if the shadow price capacity is nonlinear in the random disturbances, ε _t . We now examine the idea of improved coordination by appointing the upstream division a “gatekeeper” who must approve capacity rights secured by the other division.

Suppose that, as in Section 3, a central unit procures new capacity as needed. However, instead of having the right to secure capacity unilaterally for the current period, the downstream division can now only do so through a mutually acceptable negotiation with the upstream division.³⁰ If the two divisions reach an upfront agreement, it specifies Division 2’s capacity rights k _2t and a corresponding transfer payment TP _t that it must make to Division 1 for obtaining these rights. The parties report the outcome of this agreement (k _2t , TP _t ) to the central office, which commits to honor it as the status quo point in any subsequent renegotiations. Division 1 then secures enough capacity from the central office to meet its own capacity needs as well as fulfill its obligation to the downstream division. As before, Division 1 is charged the historical full cost of capacity under the relative practical capacity depreciation rule (i.e., c) for each unit of capacity that it acquires from the central owner. If the parties fail to reach a mutually acceptable agreement, the downstream division would have no claim on capacity in that period, though it may, of course, obtain capacity ex post through negotiation with the other division.

Since ownership of capacity assets is centralized, the divisional capacity choice problems are again separable across time periods. Therefore, a gatekeeper arrangement will attain strong goal congruence if it induces the two divisions to acquire collectively the capacity level \(k_t^o\) in each period. The proof of Proposition 7 demonstrates that in order to maximize their joint expected surplus, the divisions will agree on a particular amount of capacity level k _2t that the downstream can claim for itself in any subsequent renegotiation. Thereafter the upstream division has an incentive to acquire the optimal amount of capacity \(k_t^o\) for period t.³¹ By taking away Division 2’s unilateral right to rent capacity at some transfer price, the central office will generally make Division 2 worse off. We note, however, that this specification of the default point for the initial negotiation is of no importance for the efficiency of a gatekeeper arrangement. The same capacity level, albeit with a different transfer payment, would result if the central office stipulated that, in the absence of an agreement at the initial stage of period t, Division 2 could unilaterally rent capacity at some transfer price p _t (for instance, p _t  = c).

Our finding that a two-stage negotiation allows the divisions to achieve an efficient outcome is broadly consistent with the results in Edlin and Reichelstein (1995) and Wielenberg (2000). The main difference is that in the present setting the divisions bargain over the downstream division’s unilateral capacity rights. As observed above, the upstream division would acquire too little capacity from a firm-wide perspective, if the downstream division could not stake an initial capacity claim. On the other hand, Proposition 5 demonstrated that simply giving the downstream division the right to acquire capacity at the relevant cost, c, could result in either over- or under-investment. By appointing Division 1 a gatekeeper for Division 2’s unilateral capacity claims, the firm effectively balances the divisional rights and responsibilities so as to obtain goal congruence.

6.3 Optimal incentive contracting

The crucial feature of the cash flow specification in (25) is, of course, the assumed additive separability of divisional revenues and the managers’ productive contributions Y _it . This separability implies that capacity investments are not a source of additional contracting frictions, and therefore the optimal second-best incentive scheme does not need to balance a tradeoff between productive efficiency and higher managerial compensation. In the models of Baiman and Rajan (1995), Christensen et al. (2002), Dutta and Reichelstein (2002), Baldenius et al. (2007), and Pfeiffer and Schneider (2007) in contrast, the investment decisions are a source of informational rent for the manager, and as a consequence, optimality requires a departure from the first-best investment levels. In these papers, the second-best policy entails lower investment levels, which can be induced through a suitable increase in the “hurdle rate,” that is, the capital charge rate applied to the book value of assets. It remains an open question for future research whether similar results can be obtained in the context of multiple overlapping investment decisions.