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.
The firm seeks a path of efficient investment and capacity levels so as to maximize the stream of discounted future cash flows. Suppose hypothetically that a central planner had the entire information regarding future revenues, that is, the sequence of future θ
t
’s. The optimal investment decisions
b ≡ (
b
1,
b
2, ...) would be chosen so as to maximize the net present value of the firm’s expected future cash flows:
$$ \Uppi_{d}(b) = \sum_{t=1}^{\infty} \left[M_{d}(b_{t}+ \beta \cdot b_{t-1}, \theta_t) - v \cdot (1+r) \cdot b_t\right] \cdot \gamma^t, $$
subject to the non-negativity constraints
b
t
≥ 0. Here,
M
d
(
b
t
+ β ·
b
t−1, θ
t
) denotes the maximized value of the firm-wide contribution margin:
$$ E_{\epsilon}\left[R_1(k_{1t}, \theta_{1t}, \epsilon_{1t}) + R_2(k_{2t}, \theta_{2t}, \epsilon_{2t})\right], $$
subject to the constraint that
k
1t
+
k
2t
≤
b
t
+ β ·
b
t−1.
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
Lemma 1 also identifies
c as the effective long-run marginal cost of capacity. An intuitive argument for this characterization is that the firm can increase its capacity in period
t by one unit without affecting its capacity levels in subsequent periods through the following “reshuffling” of future capacity acquisitions: buy one more unit of capacity in period
t, buy β unit less in period
t + 1, buy β
2 more unit in period
t + 2, and so on. The cost of this variation, evaluated in terms of its present value as of date
t − 1, is given by:
$$ v \cdot \left[ 1 - \gamma \cdot \beta + \gamma^2 \cdot \beta^2 - \gamma^3 \cdot \beta^3 + \gamma^4 \cdot \beta^4 \ldots \right] = v \cdot \frac{1}{1 + \gamma \cdot \beta}, $$
and therefore the present value of the variation at date
t (i.e., the end of period
t) is:
$$ (1+r)\cdot v \cdot \frac{1}{1 + \gamma \cdot \beta} \equiv c. $$
Hence
c is the marginal cost of one unit of capacity made
available for one time period. It is useful to note that
c is exactly the price that a hypothetical supplier would charge for renting out capacity for one period, if the rental business is constrained to make zero economic profit. Accordingly, we will also refer to
c as the
competitive rental price of capacity.
13
In the context of a single division, Rogerson (
2008) has identified depreciation rules that result in goal congruence with regard to a sequence of overlapping investment projects. The depreciation schedule can be set in such a manner that the historical cost charge (the sum of depreciation and imputed interest charges) for one unit of capacity in each period is precisely equal to
c, the marginal cost of capacity derived in Lemma 1. Let
z
t-1, t denote the historical cost charge in period
t per dollar of capacity investment undertaken at date
t − 1. It consists of the first-period depreciation percentage
d and the capital charge
r applied to the initial expenditure required for one unit of capacity. Thus:
$$ z_{t-1,t} = v \cdot (d +r). $$
Accordingly,
z
t−2,t denotes the cost charge in period
t per dollar of capacity investment undertaken at date
t − 2, and:
$$ z_{t-2,t} = v \cdot \left[(1-d) + r \cdot (1-d)\right]. $$
The total historical cost charge to Division 1’s residual income measure in period
t then becomes:
$$ z_{t}\equiv z_{t-1,t} \cdot b_{t} + z_{t-2,t}\cdot b_{t-1}. $$
Division 1 will internalize a unit cost of capacity equal to the firm’s marginal cost
c, provided
$$ z_{t} = c \cdot (b_{t} + \beta \cdot b_{t-1}) = c \cdot k_t. $$
Straightforward algebra shows that there is a unique depreciation percentage
d that achieves the desired intertemporal cost allocation of investment expenditures. This value of
d is given by:
$$ d= \frac{1}{\gamma +\gamma^2\cdot\beta} - r. $$
(8)
We note that 0 <
d < 1 and:
$$ (z_{t-2, t}, z_{t-1, t}) = \left( \frac{\beta \cdot v}{\gamma +\gamma^2\cdot\beta}, \frac{v}{\gamma+\gamma^2\cdot\beta}\right)=(\beta\cdot c, c). $$
(9)
Thus the historical cost charge per unit of capacity is indeed
c in each period. The above intertemporal cost charges have been referred to as the
relative practical capacity rule since the expenditure required to acquire one unit of capacity is apportioned over the next two periods in proportion to the capacity created for that period, relative to the total discounted capacity levels.
14 We note in passing that the depreciation schedule corresponding to the relative practical capacity rule will coincide with straight line depreciation exactly when
\(\beta = \frac{1+r}{1+2r}\). For instance, if
r = 0.1, the relative practical capacity rule amounts to straight line depreciation if the practical capacity in the second period declines to 91%.
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 now relax the assumption of dedicated capacity. A plausible alternative scenario, which we maintain throughout the remainder of this paper, is that the demand shocks
\(\epsilon\)
t
are realized sufficiently early in any given period and the production processes of the two divisions have enough commonalities so that the divisional capacity uses remain
fungible. While the total capacity,
k
t
, is determined at the beginning of period
t, this resource can be reallocated following the realization of the random shocks
\(\epsilon\)
t
. To that end, we assume that the two divisions are free to negotiate an outcome that maximizes the total revenue available,
\(\sum_{i=1}^{2} R_{i}(q_{it},\theta_{it}, \epsilon_{1t})\) , subject to the capacity constraint
q
1t
+
q
2t
≤
k
t
. Provided the optimal quantities
\(q_{i}^*(k_t, \theta_t, \epsilon_t)\) are positive, they will satisfy the first-order condition:
$$ R_{1}^{'}(q_{1t}^*, \theta_{1t}, \epsilon_{1t}) = R_{2}^{'}(k_t - q_{1t}^*, \theta_{2t}, \epsilon_{2t}). $$
(10)
We also define the
shadow price of capacity in period
t, given available capacity,
k
t
, as:
$$ S(k_t, \theta_t, \epsilon_t) \equiv R_{i}^{'}(q_{i}^*(k_t, \theta_t,\epsilon_t), \theta_{it}, \epsilon_{it}), $$
(11)
provided
\(q_{i}^*(k_t,\theta_t,\epsilon_t)>0\). Thus, the shadow price is the marginal revenue that the divisions could collectively obtain from an additional unit of capacity acquired at the beginning of the period. Clearly,
S(·) is increasing in both θ
t
and
\(\epsilon\)
t
but decreasing in
k
t
.
The net present value of the firm’s expected future cash flows is now given by:
$$ \Uppi_{f}(b) = \sum_{t=1}^{\infty} E_{\epsilon} \left[M_{f}(b_{t}+ \beta \cdot b_{t-1}, \theta_t,\epsilon_t) - v \cdot b_t\right] \cdot \gamma^t, $$
where the maximized contribution margin now takes the form:
$$ M_{f}(k_t, \theta_t,\epsilon_t) = R_1(q_1^*(k_t,\theta_t,\epsilon_t), \theta_{1t}, \epsilon_{1t}) + R_2(k_t - q_1^*(k_t, \theta_t,\epsilon_t), \theta_{2t}, \epsilon_{2t}). $$
Using the Envelope Theorem, we obtain the following analogue of Lemma 1.
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.
Since the relevant information embodied in the shocks
\(\epsilon\)
t
is assumed to be known only to the divisional managers and they are assumed to have symmetric information about the attainable net revenues, the two divisions can split the “trading surplus” of
\( M_f(k_t, \theta_t, \epsilon_t) - \sum_{i=1}^{2} R_i(k_{it}, \theta_{it}, \epsilon_{it})\) between them. Let δ ∈ [0, 1] denote the fraction of the total surplus that accrues to Division 1. Thus, the parameter δ measures the relative bargaining power of Division 1, with the case of
\(\delta=\frac{1}{2}\) corresponding to the familiar Nash bargaining outcome. The negotiated adjustment in the transfer payment, Δ
TP is then given by:
$$ R_1(q_1^*(k_t,\theta_t,\epsilon_t), \theta_{1t}, \epsilon_{1t}) + \Updelta TP = R_1(k_{1t}, \theta_{1t}, \epsilon_{1t}) + \delta \cdot \left [M_f(k_t, \theta_t, \epsilon_t) - \sum_{i=1}^{2} R_i(k_{it}, \theta_{it}, \epsilon_{it}) \right]. $$
At the same time, Division 2 obtains:
$$ R_2(k_t - q_1^*(k_t, \theta_t,\epsilon_t), \theta_{2t}, \epsilon_{2t}) - \Updelta TP = R_2(k_{2t}, \theta_{2t},\epsilon_{2t}) + (1-{\delta)} \cdot \left [M_f(k_t, \theta_t, \epsilon_t) - \sum_{i=1}^{2} R_i(k_{it}, \theta_{it}, \epsilon_{it}) \right]. $$
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.