Entry Deterrence and Excess Capacity:
The Case of Concave Capacity Costs
P. van den Boogaard
Supervisor: dr. A. van der Made
University of Groningen
EBM877A20
June 6, 2018
Abstract
1
Introduction
Deterring entry by holding excess capacity has been subject to much discussion in the past. For example, Spence (1977), Bulow et al. (1985) and Poddar (2003) support the premise that holding excess capacity could deter entry. On the other hand, the effective-ness of entry deterrence by holding excess capacity has been criticised by a considerable amount of scholars. See for instance, Dixit (1980), Schmalensee (1981) and Spulber (1981).
In my thesis, I use a unique model to determine whether it is possible to deter entry by holding excess capacity. In this model, a single incumbent faces a threat of entry of a single entrant. Furthermore, the incumbent is the only firm that is able to install capacity. The cost function of the incumbent is defined to allow for concave capacity costs. Additionally, the incumbent is able to produce at a lower cost up to his capacity level. This combination of features ensures the uniqueness of my model. That is, the effectiveness of holding excess capacity to deter entry is tested in a setting that has not been considered before. The research question that is addressed in this thesis is for-mulated as follows: is deterring entry by holding excess capacity possible for a single incumbent if it faces a threat of entry?
This approach shows that entry deterrence by holding excess capacity is more likely in industries where the marginal costs are low, the reduction in costs up to the capacity level is great and the costs of capacity are relatively concave. The effect that the fixed entry cost has on the capacity decision is ambiguous.
The structure for the remainder of this thesis is as follows: in section 2, relevant papers on the subject of entry deterrence using excess capacity are discussed. In section 3, a model is constructed in order to answer the research question. In this section, each time period of the game is discussed in a separate subsection. In section 4 the implications of the model are discussed by way of an intuitive approach. In section 5, limitations of the model are reviewed. Section 6 is dedicated to the conclusion.
2
Literature Review
Eaton and Ware (1987) consider a model in which costs of capacity are concave. Addi-tionally, when these costs have been made, they are considered sunk. However, contrary to my model, Eaton and Ware allow for multiple firms that are all able to set capacities. Moreover, these firms are unable to produce above their capacity level and the model does not feature an interaction term between capacity and cost of production. The au-thors find that no firm holds excess capacity in equilibrium. Their reasoning is that eliminating excess capacity leads to an increase in net profits. The authors conclude that entry can be either blockaded or (strategically) deterred, depending on the fixed cost of entry and the sunk component of production costs. Furthermore, Spulber (1981) reviews a dynamic model of entry, whereas Dixit (1980) uses a static framework. Ad-ditionally, Spulber allows both firms to set capacity. His findings are similar to those of Dixit. Namely, Spulber finds that the incumbent generally does not hold excess ca-pacity. Exceptions are made for very specific conditions, such as relatively inexpensive capacity and demand fluctuations due to entry.1 Moreover, Schmalensee (1981) uses a similar model as the aforementioned. However, in contrast to Spulber, Schmalensee uses a model in which only the incumbent is able to set capacity. Nonetheless, his find-ing is similar; capacity is always completely utilized by the incumbent.
The main conclusion from the literature discussed in the paragraph above is that holding excess capacity to deter entry as described by Spence is ruled out. In constrast, Bulow et al. (1985) show that an incumbent is able to hold excess capacity in equilibrium. The authors note that each firm’s marginal revenue is not necessarily decreasing in the other firm’s output, which is assumed by Dixit (1980). If this assumption made by Dixit is relaxed, for instance by adopting a demand curve with constant elasticity, it is possi-ble that the incumbent deters entry while holding excess capacity. Additionally, Poddar (2003) shows that the incumbent may hold excess capacity in equilibrium when future demand is not completely certain. In this case, the incumbent either successfully deters entry, or holds a strategic advantage over the entrant in the case of entry. Furthermore,
1Due to entry, a portion of the incumbent’s customers move to the entrant. Thus, the incumbent
Zhang (1993) alters the model by assuming that the industry is labour-managed instead of capitalistic profit-maximizing. A firm operating in a labour-managed industry max-imizes surplus per worker, while a firm operating in a capitalistic profit-maximizing industry maximizes profit. Zhang’s model allows for successful entry deterrence by a (labour-managed) incumbent that holds excess capacity. See Vanek (1970) for more information regarding labour-managed economies and its implications.
To test whether holding excess capacity to deter entry occurs in reality, Lieberman (1987) conducts empirical research in 38 chemical product industries. These indus-tries are characterised by high fixed costs, economies of scale and a low number of incumbent firms. In theory, these characteristics should allow for effective entry deter-rence by firms that hold excess capacity. Indeed, Lieberman finds that firms in these product industries hold excess capacity. However, the author is only able to account this to entry deterrence purposes in three cases. Rather, most firms hold excess capacity to accomodate demand variability and for investment purposes. Additionally, Masson and Shaanan (1986) empirically test whether excess capacity is used to deter entry. Their sample consists of 26 industries. The authors find ”little evidence that firms strategi-cally add capacity to deter entry.” In fact, in a survey of evidence from empirical studies on entry and exit, Siegfried and Evans (1994) state that: ”In spite of the popularity of this idea among economists, there is no empirical evidence whatsoever that supports the deliberate use of excess capacity as a barrier to entry.”
remark to their findings. Namely, they are unable to rule out that excess capacity is installed for other reasons than to deter entry. The authors indicate that they believe that their original explanation is more reasonable, but do not provide any supporting argu-ments to this belief. Goolsbee and Syverson (2008), do not find evidence in favour of entry deterrence by holding excess capacity. In their paper, the authors use data from the U.S. airline industry to analyse how incumbents respond to threats of entry. Regarding entry deterrence by holding excess capacity, the authors note that a rise in capacity in response to a threat of entry cannot be ruled out. However, supporting evidence is weak.
As shown in this section, there has been much debate on the use of excess capacity. Scholars seem to disagree on its ability to deter entry. In general, excess capacity ap-pears to be most effective when certain conditions apply, if effective at all.2 An
examina-tion of empirical evidence demonstrates that excess capacity is not the most prominent entry deterrent. Notwithstanding, there are indications that holding excess capacity is used by firms to deter entry.
3
The Model
In the game that is discussed below, firm 1 is regarded as the incumbent and firm 2 as the potential entrant. The inverse demand function is specified as follows: p = 1−Q. In this formula, price is denoted by p and aggregate output by Q. Due to this specification of the inverse demand function, aggregate output never exceeds unity. The timing of the game is as follows:
t = 1: The incumbent decides on its capacity level t = 2: The entrant decides on entry
t = 3: If entry occurs, the firms compete in quantities.
If entry does not occur, the incumbent acts as a monopolist.
In the first period the incumbent sets a capacity level. As aggregate output never exceeds unity, it is assumed that the incumbent never installs capacity that exceeds the maximum
amount of quantity that he can produce. Since installing a negative amount of capacity is not possible either, it is assumed that 0 ≤ k1 ≤ 1. Here, k1 denotes capacity installed
by the incumbent. As the capacity decision occurs before the entry decision, the entrant is unable to set capacity in this model. In the first period, the costs of installing capacity are incurred directly. This implies that the costs of capacity are sunk in the competi-tion stage. Moreover, economies of scale are assumed in the installment of capacity. This means that the (sunk) costs of capacity are concave. Formally, the costs of the installment of capacity are defined as follows: φk1− βk12. Concavity is ensured by the
parameters φ and β. As long as β is small enough compared to φ, concavity is assured. Formally, this leads to the condition 2β ≤ φ. This condition transforms to β ≤ 12, as φ is normalized to unity and 0 ≤ k1 ≤ 1. If this condition is not imposed, the incumbent
could face decreasing production costs for large capacities.
Furthermore, the benefit of installing capacity is included in the cost function by the fol-lowing terms: wq1− γk1q1. The first term, the regular costs of production, are lowered
by the second term, which is called the interaction term. The interaction term allows the incumbent to produce at a lower cost up to his predetermined capacity level. The mag-nitude of this interaction effect is determined by γ. The larger γ, the stronger this effect. If γ is equal to zero, the interaction effect disappears. The interaction effect can be justified as follows: an incumbent that has installed capacity prior to production, is able to produce more efficiently up to this level than above it. Production up to the capacity level is anticipated by the incumbent, which implies that provisions have been made in order to let production run smoothly. Contrarily, if the incumbent produces above the predetermined capacity level, it has to take unanticipated measures. An example could be that production inputs need to be acquired almost instantly, which implies that they have to be secured at a premium. Therefore, the interaction effect captures the discount that the incumbent (implicitly) receives from producing up to the capacity level.
planes on a relatively short notice. If the deadline is exceeded, the aerospace manu-facturer has to compensate the airline for losses of income. As firm A holds excess capacity, it is likely that this firm is able to produce the planes at a lower cost. First of all, this firm can start production almost instantly. This means that firm A is more likely to meet the delivery date than firm B. Hence, firm A is less likely to incur the penalty. Additionally, since production in the aerospace industry is relatively specific, it is expected that the supply of production inputs is not perfectly efficient. In line with the paragraph above, this means that firm B needs to acquire these inputs at a premium. Namely, firm B needs to secure these inputs almost instantly. This is not the case for firm A, as this firm holds excess capacity. These two characteristics imply that hold-ing excess capacity could potentially lower the cost of production. More generally, the interaction effect is more likely to be evident in industries that feature characteristics similar to the aerospace industry. On the other hand, it is not very likely that the in-teraction effect is visible in industries where late delivery is not severely punished and production inputs are widely available and can be swiftly obtained.
In the second period, the entrant decides on entry. If entry occurs, the entrant has to incur a fixed entry cost equal to F . Subsequently, in the third period, actual compe-tition takes place. The regular marginal cost of production is identical for both firms and denoted by w. Naturally, regular marginal costs of production must be nonnegative. Also, regular marginal costs of production should be lower than unity. Namely, when w ≥ 1, the active firms are unable to produce a positive amount of aggregate output in equilibrium. Hence, 0 ≤ w < 1. The aforementioned is displayed in the cost functions below: C1 = wq1− γk1q1+ φk1− βk12 f or q1 ≤ k1 wk1− γk12 | {z } (i) + w(q1 − k1) | {z } (ii) + φk1− βk21 | {z } (iii) f or q1 > k1 (1) C2 = wq2+ F (2)
that γ < w. This means that the reduction in cost due to the interaction effect can never be larger than the marginal cost of production in itself. Furthermore, the incumbent faces a sunk cost of capacity.
On the other hand, when the incumbent produces above his capacity level, the sec-ond line of (1) applies. The first part, denoted by (i), are the costs that the incumbent faces from producing up to his capacity level. The second part, denoted by (ii), are the costs that the incumbent faces for producing above this capacity level. Note that for production above the capacity level, the incumbent can no longer benefit from the cost reduction through the interaction term. The last part, denoted by (iii), are the sunk costs of installing the predetermined level of capacity.
Note that the cost function of the entrant, as described in (2), does not feature an in-teraction term, nor sunk costs of installing capacity, as the entrant is unable to install any capacity nor benefit from it. However, the cost function of the entrant does include a fixed cost of entry.
Below, the game is solved by backward induction. To clearly distinguish between the different periods in the game, each period is discussed in a separate subsection. The third period, in which the Cournot profits of the two firms are derived in case of entry, is discussed in subsection 3.1. This subsection is split into two parts: in the first part, the incumbent holds excess capacity. In the second part, the incumbent does not hold excess capacity. In the second period, the entrant decides upon entry. This time period of the game is discussed in subsection 3.2. In the first period, the incumbent decides on its capacity level, which is discussed in subsection 3.3.
3.1
Cournot competition
3.1.1 With excess capacity
In this case, the net profit functions of the two firms look as follows:
π1 = pq1− C1 = (1 − q1− q2)q1− wq1+ γk1q1− k1+ βk12 (3)
π2 = pq2− C2 = (1 − q1− q2)q2− wq2− F (4)
Taking the first-order conditions yields:
∂π1 ∂q1 = 0 → 1 − 2q1− q2− w + γk1 = 0 (5) 2 ∗∂π2 ∂q2 = 0 → 2 − 2q1− 4q2− 2w = 0 (6)
Subtracting (6) from (5) leads to the equilibrium output of the entrant:
q2∗ = 1 − w − γk1
3 (7)
Substituting (7) into (5) gives the equilibrium output of the incumbent:
q∗1 = 1 − w + 2γk1
3 (8)
Therefore, total output by the industry equals:
Q∗ = 2 − 2w + γk1
3 (9)
and the equilibrium price equals:
p∗ = 1 + 2w − γk1
3 (10)
Ultimately, the equilibrium net profits for both firms are calculated:
π∗2 = (p∗− w)q∗ 2 − F = 1 − w − γk1 3 2 − F π∗2 = 19(1 − w − γk1)2 − F (12)
The equilibrium net profits of the incumbent as described in (11) are decreasing in the regular marginal costs of production. This is in turn, through the interaction effect, mit-igated by his capacity level. Additionally, the sunk costs of capacity have a negative impact on the equilibrium net profits of the incumbent. Note that (11) depicts the equi-librium net profits in the case where the incumbent does hold excess capacity. By defi-nition, this means that q1∗ ≤ k1. This condition is examined in more detail in subsections
3.3.1 and 3.3.2. Furthermore, the equilibrium net profits of the entrant, as described in (12), are decreasing in the regular marginal costs of production. Also, the equilibrium net profits of the entrant are decreasing in the capacity level of the incumbent. This is logical, as the interaction effect allows his competitor to have lower production costs. Hence, the profits that the entrant can earn decrease when this effect becomes stronger. Furthermore, the fixed entry cost has a negative impact on the equilibrium net profits of the entrant.
3.1.2 Without excess capacity
The derivation of the equilibrium profits without excess capacity is shown below.
π1 = pq1− C1 = (1 − q1− q2)q1− wk1+ γk12− w(q1− k1) − k1+ βk21 (13)
π2 = pq2− C2 = (1 − q1− q2)q2− wq2− F (14)
Subsequently, taking the first-order conditions yields:
∂π1 ∂q1 = 0 → 1 − 2q1− q2− w = 0 (15) ∂π2 ∂q2 = 0 → 1 − q1− 2q2− w = 0 (16)
Invoking symmetry (q1 = q2 = ˆq) yields the equilibrium outputs of the two firms:
ˆ
q = ˆq1 = ˆq2 =
1 − w
Therefore, total output by the industry equals:
ˆ
Q = 2(1 − w)
3 (18)
and the equilibrium price equals:
ˆ
p = 1 + 2w
3 (19)
Ultimately, the equilibrium net profits for both firms are calculated:
ˆ π1 = ˆpˆq1− wk1+ γk12− w(ˆq1 − k1) − k1+ βk12 ˆ π1 = (ˆp − w)ˆq1+ γk21− k1+ βk12 ˆ π1 = 19(1 − w)2+ γk21 − k1+ βk12 (20) ˆ π2 = ˆpˆq2− w ˆq2− F = 19(1 − w)2 − F (21)
The equilibrium net profits of the incumbent as described in (20) are decreasing in the regular marginal costs of production. As the incumbent is only able to benefit from the interaction effect for his production up to the predetermined capacity level, the gain from this effect only extends to k1. Note that (20) depicts equilibrium net profits in
the case where the incumbent does not hold excess capacity. By definition, this means that ˆq1 > k1. This condition is examined later in this thesis, specifically in subsection
3.2
Entry decision
In this subsection, the entry decision is elaborated upon. Entry occurs when the entrant earns nonnegative net profits. So when the incumbent is going to hold excess capacity, entry occurs when:
π2∗ ≥ 0 → 1
9(1 − w − γk1)
2 ≥ F (22)
and when the incumbent is not going to hold excess capacity, entry occurs when:
ˆ
π2 ≥ 0 → 19(1 − w)2 ≥ F (23)
or:
1 − w ≥ 3√F (24)
3.3
Capacity decision
The incumbent has three options to set capacity. Firstly, the incumbent could deter entry and consequently earn monopoly profits. Secondly, the incumbent could accommodate entry while holding excess capacity. Thirdly, the incumbent could accommodate entry without holding excess capacity. Naturally, the incumbent sets capacity according to the scenario that yields the highest net profits. Below, the net profits of the three scenarios are calculated in separate subsections. In the final subsection, the equilibrium net profits of the incumbent are compared, in order to determine the capacity decision.
3.3.1 The incumbent deters entry
In order to deter entry, the incumbent has to ensure that the entrant is unable to earn a nonnegative profit. So:
π2∗ < 0 → 19(1 − w − γk1)2− F < 0 (25)
From this follows:
k1 >
1
γ(1 − w − 3 √
F ) (26)
By setting a capacity level that is marginally larger than the right-hand side of (26), the incumbent is able to deter entry. I call this the minimum capacity level that ensures entry deterrence:
k1D = 1
γ(1 − w − 3 √
F ) (27)
Note that an increase in the magnitude of the interaction effect lowers the minimum capacity level that ensures entry deterrence. This is intuitive, as the interaction effect determines how much the costs of production are lowered. If this effect is strong, then a low level of capacity already ensures a significant drop in costs for the incumbent. On the contrary, if the effect is weak, a larger capacity level is needed to guarantee the same drop in costs.
Recall that 0 ≤ k1 ≤ 1. Therefore, if k1D > 1, the incumbent is never able to
de-ter entry. As my aim is to develop a model in which entry dede-terrence is theoretically possible, I assume kD1 ≤ 1. From this follows:
1
γ(1 − w − 3 √
F ) ≤ 1 (28)
Solving for γ yields:
γ ≥ 1 − w − 3√F (29)
produces when he successfully deters entry. Nevertheless, the condition still needs to hold, to ensure that the equilibrium is subgame perfect. So:
q1∗ ≤ kD 1 (30) Substitution yields: 1 − w + 2(1 − w − 3√F ) 3 ≤ 1 γ(1 − w − 3 √ F ) (31)
Solving this inequality produces the following condition:
1 − w ≥ 3 − 2γ 1 − γ
√
F (32)
Having established these two conditions, I proceed with the derivation of the profits for the incumbent when he successfully deters entry. In this case, the incumbent earns monopoly profits. The net profits of the incumbent look as follows:
π1D = pQ − C1 = (1 − Q)Q − wQ + γk1Q − k1+ βk21 (33)
Taking the first-order condition yields:
∂πD 1
∂Q = 0 → 1 − 2Q − w + γk1 = 0 (34) Solving (34) for quantity yields:
QD = 1 − w + γk1
2 (35)
As the capacity level can now be taken as given, I substitute (27) into (35). This yields:
QD = 2 − 2w − 3 √
F
2 (36)
The condition QD1 ≤ kD
1 has to hold, as the incumbent must hold excess capacity. From
Solving this inequality leads to the following condition:
1 − w ≥ 3(2 − γ) 2(1 − γ)
√
F (38)
Comparison of (32) and (38) shows that the latter condition is more stringent, as the right-hand side of (38) is strictly larger than the right-hand side of (32). This suggests that the quantity of the incumbent is larger when he deters entry, than his quantity when he accommodates entry while holding excess capacity. This has been checked and holds when capacity is set corresponding to (27).
In this thesis it is assumed that condition (38) holds. In the case that (38) does not hold, but (32) does, the incumbent deters entry without excess capacity. As this only occurs when specific parameter values are chosen, I do not consider this case in more detail in this section. However, I derive the profits of this case in the Appendix.
To derive the equilibrium price, I substitute (35) in the inverse demand function:
pD = 1 + w − γk1
2 (39)
Again, as the capacity level can now be taken as given, I substitute (27) into (39):
pD = w + 3 √
F
2 (40)
Ultimately, the equilibrium net profits for the monopolist can be calculated. Note that for this calculation, I used (35) and (39), to avoid computational errors. Afterwards, (27) is substituted into the equilibrium monopoly profits.
π1D = pDQD − wQD + γk
1QD− k1+ βk12
πD1 = 14(1 − w + γk1)2− k1+ βk21 (41)
Substituting (27) into (41) yields:
3.3.2 The incumbent accommodates entry - with excess capacity
In this scenario, the incumbent holds excess capacity and takes entry as given. Consid-ering this, he sets an optimal amount of capacity. The profit function of the incumbent when he holds excess capacity is defined in (11). The second-order derivative of this function with respect to capacity is positive, implying that this function is convex in capacity. Consequently, the optimal capacity decision is a corner solution. Either the incumbent chooses the minimum or maximum amount of capacity. The upper limit of capacity is reached by setting capacity equal to unity. In the computations below, this capacity level is denoted by ¯kA1
1 and associated profits by ¯π A1
1 . By setting capacity equal
to unity, the incumbent always holds excess capacity, as quantity never exceeds unity. The lower limit of capacity reached by setting capacity just marginally higher than out-put. This capacity level is denoted by kA1
1 and associated profits by π A1
1 . Below these
profit levels are calculated.
By taking (11) and setting k1 = ¯kA11= 1, the profits of the incumbent are calculated
when he holds excess capacity and sets capacity equal to the upper limit:
¯ πA1 1 = 1 9(1 − w + 2γ) 2− 1 + β (43)
In equilibrium, the incumbent must earn nonnegative profits. Therefore:
1
9(1 − w + 2γ)
2+ β ≥ 1 (44)
Analysis of (44) shows that this inequality is never satisfied. Notice that the left-hand side of (44) is positively related to γ and negatively related to w. Since the condition γ < w has to be taken into account, the left-hand side is maximised when γ and w are (approximately) equal to each other. In this case, (44) transforms to:
1
9(1 + w)
2+ β ≥ 1 (45)
Since β ≤ 12 and 0 ≤ w < 1, the inequality defined in (45) is never satisfied, meaning that ¯πA1
1 is always negative. Therefore, accommodating entry by holding the upper limit
Because by assumption kD
1 ≤ 1, ¯k1A1 is actually greater or equal to k1D. Therefore,
if ¯kA1
1 is installed, entry is actually be deterred. This means that in reality, the upper
limit of capacity when entry is accommodated with excess capacity is lower than unity. Recall that the associated profit function of the incumbent is convex in capacity. There-fore, profits of the incumbent are larger when capacity equals unity, than when the actual upper limit (which is lower) is installed. Since ¯πA1
1 is always negative, the incumbent
earns negative profits as well when he installs the actual upper limit of capacity.
Below, the profits of the incumbent are computed when he opts for the lower limit of capacity. In this case, k1 = kA11 = q∗1. Solving this leads to k
A1
1 = 3−2γ1−w. Substituting
this in (11) leads to:
πA1 1 = 1 9 3(1 − w) 3 − 2γ 2 − 1 − w 3 − 2γ + β 1 − w 3 − 2γ 2 (46)
In order to attain a feasible solution, πA1
1 should be nonnegative. From this follows:
1 9(3k A1 1 )2− k A1 1 + βk A1 1 2 ≥ 0 (47) Note that kA1 1 = 1−w
3−2γ is substituted back in (47). Rewriting (47) yields:
kA1 1 (k A1 1 − 1 + βk A1 1 ) ≥ 0 (48)
Solving this equation for kA1
1 yields the following conditions:
kA1 1 ≤ 0 ∨ k A1 1 ≥ 1 1 + β (49)
Note that in this case the incumbent holds excess capacity. Therefore, the following condition has to hold: 0 < kA1
1 ≤ 1. Consequently, k A1
1 ≤ 0 is never satisfied. The
second inequality of (49) is solved below. In this case, I assume that β = 12, as this minimizes the right-hand side of the second inequality. Hence, the range for which this inequality is satisfied is made as large as possible.
kA1
1 ≥
2
Substituting kA1 1 = 1−w 3−2γ yields: 1 − w 3 − 2γ ≥ 2 3 (51)
This inequality is solved using software.3 This yields:
3
4 < γ < 1 ∧ 0 ≤ w ≤ (4γ − 3) (52) The solution in (52) is never satisfied, as γ is not allowed to be larger than w. Therefore, both conditions described in (49) never hold. This implies that πA1
1 is always negative
and consequently kA1
1 is never chosen by the incumbent.
This subsection demonstrates that it is never possible for the incumbent to accommo-date entry with excess capacity. The profits of the incumbent are strictly negative in both potential optima. This implies that ¯kA1
1 and k A1
1 are never chosen by the incumbent.
Consequently, a comparison of the associated profits is not included in this section.
3.3.3 The incumbent accommodates entry - without excess capacity
Similar to the section above, the profit function of the incumbent when he does not hold excess capacity, as described in (20), is convex in capacity. Therefore, the optima are found in the same manner. I first formulate the upper and lower limit of capacity. The upper limit of capacity is denoted by ¯kA2
1 and the lower limit by k A2
1 . In this section,
the incumbent does not hold excess capacity. When the incumbent opts for the upper limit, this is ensured when ¯kA2
1 < ˆq1. Hence, the upper limit of capacity is reached by
setting capacity marginally lower than his output, as described in (17). Furthermore, the lower limit of capacity is reached by setting capacity equal to zero. Below the associated profits are computed and compared. Naturally, the incumbent opts for the capacity level that yields the highest profit.
By taking (20) and setting k1 = ¯kA12 = ˆq1 = 1−w3 , the profits of the incumbent are
calculated when he does not hold excess capacity and sets capacity equal to the upper limit: ¯ πA2 1 = 1 9(1 − w) 2+ γ 1 − w 3 2 − 1 − w 3 + β 1 − w 3 2 (53) In equilibrium, the incumbent must earn a nonnegative profit. Therefore:
1
9(1 + γ + β)(1 − w) ≥ 1
3 (54)
Using the same argument as used in (44), the left-hand side is maximised when γ and w are (approximately) equal to each other. In this case:
(1 + w + β)(1 − w) ≥ 3 (55)
Since β ≤ 12 and 0 ≤ w < 1, the inequality defined in (55) is never satisfied, meaning that ¯πA2
1 is strictly negative. Therefore, ¯k A2
1 is never chosen by the incumbent.
Below, the profits of the incumbent are computed when he opts for the lower limit of capacity. In this case, k1 = kA12 = 0. Substitution reduces (20) to the following
equation: πA2 1 = 1 9(1 − w) 2 (56)
Naturally, (56) is nonnegative for the whole interval of w. Since ¯πA2
1 < 0 and π A2
1 > 0, the
incumbent always opts for kA2
1 when he accommodates entry and does not hold excess
capacity. That is, the incumbent refrains from installing capacity in this case.
3.3.4 Comparison of the three scenarios
As shown in the three subsections above, the incumbent is left with two options to set capacity. To sum up, the incumbent could set kD1 to deter entry. As a consequence, he earns corresponding profits that are denoted by πD1 . When the incumbent decides to accommodate entry while not holding excess capacity, he sets kA2
1 . The related profits
are denoted by πA2
1 . Naturally, the incumbent sets capacity according to the dominating
profit level. Formally, this means that kD
1 is installed when πD1 ≥ π A2
1 . Otherwise, the
I have attempted to solve πD 1 ≥ π
A2
1 . The calculations are included in the Appendix. In
order to solve this inequality, I assume values for β and γ. Nevertheless, I am unable to complete the calculations. In spite of this, one could question the relevance of solving the model of this thesis while assuming values for at least three parameters (γ, β and φ). This means that a solution is only applicable to an industry that matches this unique case. Therefore, I continue by intuitively evaluating the variables and parameters in the model. This is done in the next section.
Before doing so, I first outline the capacity decision of the incumbent using figure 1. This figure may prove to be helpful, as it concisely summarises the options of the in-cumbent. As shown in figure 1, the incumbent either sets capacity equal to zero or equal to kD
1 . As blockaded entry is assumed away, entry is either accommodated or deterred
by the incumbent. From (26) follows that when k < kD
1 , entry is accommodated. In
this case, the incumbent sets k = kA2
1 = 0, as conform to subsection 3.3.3. On the other
hand, for values of k1D ≤ k ≤ 1, entry is deterred. If the incumbent deters entry, the incumbent installs k = k1D. Notice that kD1 ≤ 1, as assumed in subsection 3.3.1.
4
Implications
In this section I discuss how changes in variables and parameters affect the capacity decision of the incumbent. Specifically, the regular marginal cost of production, the fixed cost of entry, the magnitude of the interaction effect and the concavity of the cost of capacity are considered.
4.1
The regular marginal cost of production
When the incumbent accommodates entry, no capacity is installed and the incumbent is unable to take advantage of the interaction effect. Hence, his marginal costs of pro-duction equal w in this case. On the other hand, when the incumbent deters entry, at least some capacity is installed. Therefore, the incumbent is able to profit from the inter-action effect. Consequently, his marginal costs of production equal w −γk1Din this case.
The ratio w−γkw D 1
determines how much these two marginal costs differ. The larger this ratio, the greater the difference between the two marginal costs. That is, for larger ratios, the incumbent faces larger reductions in costs when he deters entry. Notice that:
∂ ∂w w w − γkD 1 = −γk D 1 (w − γkD 1 )2 < 0 (57)
From (57) it is evident that increases in w, the regular marginal cost of production, reduce this ratio. Therefore, the difference between the marginal costs when entry is deterred and the marginal costs when entry is accommodated is less distinct for larger values of w. So, for lower values of w the difference between the two marginal costs is greater. Hence, deterring entry is more attractive when the regular marginal costs of production are low.
4.2
The fixed cost of entry
from the fact that the fixed cost of entry has a negative effect on the profit of the en-trant. So, a lower fixed cost of entry means that the entrant is able to earn larger profits. Consequently, entry is more likely when the associated fixed cost is low. Note that the capacity level of the incumbent has a negative effect on the profits of the entrant. So for a relatively low value for F , the incumbent needs to install more capacity to ensure that the entrant’s potential profits are negative. Effectively, the lower F , the larger that k1D has to be in order to deter entry. The effect that this has on πD1 is ambiguous. That is, the effect of an increase in k1D on π1D could be both positive and negative, depending on the parameter values. For instance, installing additional capacity could increase πD1 when the sunk costs of installing capacity are concave to a high degree (large β). Also, a strong interaction effect (large γ) could cause kD
1 to have a positive effect on π1D. On
the other hand, when the sunk costs of installing capacity are perfectly linear, and the interaction effect is nonexistent, an increase in kD
1 depresses πD1 .
So, to sum up, a lower fixed cost of entry causes an increase in the minimum amount of capacity needed to deter entry. This could either increase or decrease π1D, depending on the initial values of γ and β. Therefore, I am unable to draw an unambiguous conclu-sion from a decrease in F . For high initial values of γ and β, it may increase π1D, and consequently make entry deterrence more attractive. Contrarily, for low initial values of γ and β, it may depress πD1 and consequently make entry deterrence less attractive.
4.3
The magnitude of the interaction effect
4.4
The concavity of the cost of capacity
Larger values for β imply more concave costs of installing capacity when compared to low values of β. Since the incumbent does not install capacity when he accommodates entry, changes in β have no effect on the profits of the incumbent when he accommo-dates entry. However, capacity is installed when the incumbent deters entry. A more concave cost function, caused by an increase in β, means that the incumbent can ben-efit more from economies of scale when installing capacity. Consequently, costs are expected to decrease and as a result π1D is expected to increase. This reduction in costs (and increase in πD
1 ) is likely to be steeper in specifications where kD1 is large. As in this
case, the benefits from economies of scale are larger than for low values of kD 1 .
To sum up, an increase in β does not affect πA2
1 and increase π1D. This suggests that
when β is large, the incumbent is more likely to prefer k1D over kA2
1 . Therefore,
in-creases in β encourage entry deterrence.
5
Discussion
I am unable to give a decisive answer to the research question, as the model is not solved in its entirety. Also, an attempt to solve the model by choosing realistic parameter values did not yield a satisfactory solution. This means that the implications of the model are purely intuitive and cannot be formally proven. Therefore, future research is necessary to decisively answer the research question. Also, if future research is able to completely solve the model, one could verify the accurateness of the intuitive implications made in this thesis.
cost of entry is trivial and output is significantly larger in case of entry, the reverse may hold. From this example follows that the identification of entry deterrence holds little value, without knowing if entry deterrence is welfare improving or deteriorating. If fu-ture research finds a solution to the model of this thesis, I suggest to perform a welfare analysis. This way, the competition authority can determine whether preventive actions against entry deterrence are warranted.
6
Conclusion
In this thesis I present a model that is used to assess the possibility of an equilibrium in which entry is deterred by holding excess capacity. This model is not solved completely in this paper. Therefore, I am not able to provide a formal answer to the research ques-tion. However, there are indications that equilibria exist where entry is deterred by an incumbent that holds excess capacity. By using an intuitive approach, I indicate when such equilibria are more likely to occur.
Firstly, entry deterrence by holding excess capacity is more likely when the regular marginal costs of production are relatively low. Secondly, it is more probable that entry is deterred when the costs of installing capacity are relatively concave. Namely, when the costs of installing capacity are relatively concave, installing capacity is less costly than when the costs are relatively linear. Thirdly, a relatively strong interaction ef-fect implies that installed capacity significantly lowers the marginal cost of production. Therefore, it is more likely that entry is deterred when the interaction effect is relatively strong.
Appendix
Analysis of entry deterrence without excess capacity
When (38) does not hold, but (32) does, then 3−2γ1−γ√F < 1 − w < 3(2−γ)2(1−γ)√F . Conse-quently, the incumbent deters entry without holding excess capacity. In this case, the profits of the incumbent are equal to:
ˇ π1 = pQ − C1 = (1 − Q)Q − wkD1 + γk D 1 2 − w(Q − kD1 ) − kD1 + βk1D2 (A.1)
Note that in (A.1) the second line of (1) is used, as the incumbent does not hold excess capacity. Capacity kD
1 ensures that the incumbent is the only active firm. The optimal
level of output is found by taking the first-order condition:
∂ ˇπ1
∂Q = 0 → 1 − 2Q − w = 0 (A.2) Solving (A.2) for quantity yields:
ˇ
Q = 1 − w
2 (A.3)
Consequently, the equilibrium price equals:
ˇ
p = 1 + w
2 (A.4)
Next, the equilibrium net profits for the incumbent are calculated:
ˇ π1 = ˇp ˇQ − wk1D+ γk1D 2 − w( ˇQ − k1D) − k1D+ βk1D2 ˇ π1 = 1 4(1 − w) 2− kD 1 [1 − (γ + β)k D 1 ] (A.5)
Substituting (27) into (A.5) yields:
ˇ π1 = 1 4(1 − w) 2− 1 γ(1 − w − 3 √ F )[1 − 1 γ(γ + β)(1 − w − 3 √ F )] (A.6)
Comparison of πD
1 and π A2
1
Below, an attempt to solve the inequality πD 1 ≥ π
A2
1 is displayed. This is done in
order to determine the optimal capacity decision of the incumbent. However, the profit functions feature too many variables and parameters to attain an interpretable outcome. Therefore, I set γ = 17 and β = 12 in the calculations below. By taking these values, both profit functions of the incumbent are positive for at least some interval of w > γ. This guarantees the relevance of the comparison of the profit levels.
πD1 ≥ πA2
1 (A.7)
Substitution of (42) and (56) while accounting for γ = 17 and β = 12 yields:
1 4(2 − 2w − 3 √ F )2− 7(1 − w − 3√F ) + 49 2 (1 − w − 3 √ F )2 ≥ 1 9(1 − w) 2 (A.8)
Setting x = 1 − w − 3√F and m = 1 − w yields:
1 4(m + x) 2− 7x + 49 2 x 2 ≥ 1 9m 2 (A.9) Moving terms to the left-hand side and replacing the inequality sign allows for the use of the quadratic formula. Additionally, both sides are multiplied by the least common multiple, 36:
891x2+ 18(m − 14)x + 5m2 = 0 (A.10) Using the quadratic formula, this equation is solved for x. Note that I skip the interme-diate steps.
x = 1 99
14 − m ±√2√98 + 14m − 27m2 (A.11)
Substitution of x = 1 − w − 3√F and m = 1 − w yields:
1 − w − 3√F = 1 99
Ideally, I would like to solve the equation above for w. This way, it would be possible to determine the dominating profit level for ranges of w. However, I am not able to solve (A.12) for w. Therefore, (A.12) is solved for w using software.4 This yields:
w = 1 941 3 q 6 q 294 − 729F − 196√F − 2700 q F + 788 (A.13) and w = 1 941 −3 q 6 q 294 − 729F − 196√F − 2700 q F + 788 (A.14)
It is important to note that (A.13) and (A.14) only yield real solutions when:
294 − 729F − 196√F ≥ 0 (A.15) Solving for F gives the following range for which (A.13) and (A.14) are defined:
0 ≤ F ≤ 98(2383 − 14 √
4570)
531441 (A.16)
(By approximation, this equals 0 ≤ F < 0.264911)
As (A.13) and (A.14) are computed using software, I do not know how to determine the correct inequality signs. In other words, I am not sure for what values of w and F , π1D is larger or smaller than πA2
1 . As π1D ≥ π A2
1 is not solved completely, I am unable to
determine the capacity decision of the incumbent.
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