Download A THREE-STAGE EQUILIBRIUM WITH SUNK COSTS

IJBPE
Volume 5 • Number 1 • January-June 2012
ISSN: 0973-5801; pp. 81-84
A THREE-STAGE EQUILIBRIUMWITH SUNK COSTS
Kazuhiro Ohnishi*
Abstract: Previous papers regarding strategic commitments with sunk costs include Dixit (1980)
on a two-stage model andWare (1984) on a three-stage model.This paper proposes an equilibrium
for the three-stage model with sunk costs.
Keywords: Sunk costs; Three-stage equilibrium
JEL classification: C72, D21, L13.
1. INTRODUCTION
This Paper considers irreversible behavior, such as a commitment to capacity, i.e.,
installing production facilities, as a strategic policy in duopolistic firms. This idea was
presented in a two-stage model by Dixit (1980) and was extended to a three-stage model
by Ware (1984). In this paper, we examine Ware’s three-stage model.
Now, suppose that a firm can adopt one of two strategic choices in which its own
profits are equal, but its rival’s profits are not equal, i.e., one with a large profit for its
rival and the other with a small profit for its rival. Which choice will the firm make? If the
firm does not collude with its rival, then the firm will make the choice that makes its
rival’s profit the lowest. We obtain a new equilibrium for the three-stage model with
sunk costs by discussing this problem.
The purpose of this study is to obtain an equilibrium for the three-stage model with
sunk costs by a new criterion.
This paper is organized as follows. In Section 2, we formulate the three-stage model.
The equilibrium of the model is discussed in Section 3. Finally, Section 4 contains
concluding remarks.
2. THE MODEL
This section formulates Ware’s three-stage model. The model assumes two firms,
firm 1 and firm 2. For the remainder of this paper, when i and j are used to refer to firms
in an expression, they should be understood to run from 1 to 2 with i ≠ j. The subscript
i(i = 1,2) denotes firm i. There is no possibility of entry or exit.
Consider an industry producing a homogeneous product with the following linear
inverse demand function:
*
Osaka University and Institute for Basic Economic Science, E-mail: [email protected]
82

Kazuhiro Ohnishi
P(q1, q2) = a – b(q1 + q2),
(1)
where a, b > 0, (q1 + q2) < a/b, and P is price per unit.
The three stages in this model are as follows. In the first stage, firm 1 chooses capacity
k1 and firm 2 observes the value of k1. In the second stage, firm 2 chooses its own
capacity k2 and firm 1 observes the value of k2. In the third stage, firms 1 and 2 decide
their outputs (q1 and q2) simultaneously and independently.
Therefore, firm i’s cost function is
if qi ≤ ki ,
αq + (1 − α )ki + fi
Ci ( qi , ki ) =  i
(2)
if qi > ki ,
 qi + f i
where α ∈ [0, 1] stands for a parameter that determines firm i’s variable cost, and fi firm
i’s fixed cost. Since we do not consider entry or exit in this paper, we omit firm i’s fixed
cost fi. Firm i’s profit is
(a − b( qi + q j ))qi − ( αqi + (1 − α )ki )
if qi ≤ ki ,
Π i ( qi , q j , ki ) = 
(3)
if qi > ki .
 (a − b( qi + q j ))qi − qi
Given qj, firm i maximizes its profit with respect to qi. If firm i’s marginal cost is
constantly equal to α, then its reaction function is defined by
Riα ( q j ) = arg max(( a − b( qi + q j ))qi − αqi )
{ qi ≥ 0}
(4)
If firm i wishes to produce greater output, it acquires additional capacity. If firm i’s
marginal cost is constantly equal to 1, then its reaction function is defined by
Ri1( q j ) = arg max((a − b( qi + q j ))qi − qi ) .
{ qi ≥ 0}
(5)
Therefore, if firm i executes the prior commitment of ki, its reaction function is
shown as follows:
 Riα ( q j )

Ri ( q j ) =  ki
 R1( q )
 i j
if qi < ki ,
if qi = ki ,
if qi > ki .
(6)
Firm i chooses a strategy that maximizes its own profit given the strategies of firm j.
3. EQUILIBRIUM OUTCOMES
In this section, we show the equilibrium of the three-stage model with sunk costs.
Both firms’ reaction curves and firm 2’s isoprofit curves are illustrated in Figure 1. In
this figure, the horizontal axis q1 is firm 1’s output, the vertical axis q2 is firm 2’s output,
MiM′i is firm i’s reaction curve when α = 1, NiN′i is firm i’s reaction curve when α ∈ [0,
1), and π2π2′ and π2′′ π2′′′ are firm 2’s isoprofit curves when α = 1. Furthermore, π2 π2′ is
A Three-Stage Equilibrium with Sunk Costs

83
firm 2’s isoprofit curve extending through the Stackelberg point S where firm 1 is the
leader and firm 2 is the follower, π2′′ π2′′′ is firm 2’s isoprofit curve extending through
the point U, and U is the intersection point of curves N1N1′ and N2N2′. If π2π2′ neither
intersects nor contacts the segment UX, then firm 1 installs capacity up to the level of S
in the first stage and sets its reaction curve to the kinked line N1URXSM1′. Firm 2’s
profit is highest at S on N1URXSM1′. Therefore, firm 2 installs capacity up to S in the
second stage, and the equilibrium occurs at the point S.1,2
However, what happens to the equilibrium in the three-stage model when π2π2′
intersects UX as is shown in this figure? If firm 1 sets its reaction curve to N1URXSM1′
by installing capacity up to the level of S in the first stage, then firm2’s profit is highest
at U on N1URXSM1′. Therefore, firm 2 will install capacity up to U and the equilibrium
will occur at U. However, firm 1’s profit is higher anywhere on the segment TS than U.3
Therefore the equilibrium at U is not preferable for firm 1. How does this situation
change when firm 1 installs capacity up to D where M2M2′ and π2′′π2′′′ intersect? Since
firm 2’s profit is D = U, there is a possibility that firm 2 will install capacity up to U in the
second stage. That is, the equilibrium may occur at U rather than D.
Then firm 1 operates at the point D′, which is slightly to the left of D on M2M2′.4
Firm 1 installs capacity up to D′ and transforms its reaction curve to the kinked bold
line N1URD′ZM1′. The comparison of firm 2’s profits is D′ > D = U. Since firm 2’s profit
is highest at D′ on URD′ where firm 2 can choose, there is no possibility for firm 2 to
choose U according to the profit maximization behavior. Thus the equilibrium occurs at
D′ in the three-stage model.
Figure 1: The Equilibrium Occurs at Point D’
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Kazuhiro Ohnishi
4. CONCLUDING REMARKS
We have examined the equilibrium of the three-stage model with sunk costs. Applying
the idea of this study to the United States of America, firm 2 would earn a slightly higher
profit, for instance one cent, at D′ than it would at D. In reality, firms will not change
their business strategies for a profit as low as one cent. If the difference in firm 2’s profits
between D and D′ is much smaller, firm 2 may attack firm 1 by choosing not D′ but U.
However, the equilibrium that appeared in this paper is theoretically rational, and it can
be applied to the real behavior of firms if the difference in firm 2’s profits between D and
D′ becomes larger.
Finally, we briefly discuss the credibility of threat in the model of this paper. Suppose
that firm 2 can tell firm 1 at the beginning of the first stage that when firm 1 installs
capacity up to the level of D in the first stage, firm 2 will install capacity up to the level
of U in the second stage. At this time, because firm 2’s profits are indifferent between D
and U, this threat is credible. Thus, the equilibrium also occurs at D′.
NOTES
1.
The point S is set as the Stackelberg point where firm 1 is the leader and firm 2 is the
follower. Of course, the Stackelberg point is not necessarily required. Even if the point differs
from the Stackelberg point, the equilibrium in this paper is obtained.
2.
Firm 1’s outputs at points S and M1′ are both (a–1)/2b.
3.
This is proven as follows. Firm 1’s profit at S is (a–1)2/8b. The further the point on M2M2′
gets from S, the more firm 1’s profit falls. Furthermore, firm 1’s profit is (a–1)2/9b at T and is
(a–3 + 2α) (a–α)/9b at U. Thus the comparison of firm 1’s profits is S > T > U.
4.
Applying this idea to the United States of America, we can assume that firm 2 would earn a
slightly higher profit, for instance one cent, at D′ than it would at D.
REFERENCES
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Economics, Vol. 10, pp. 20-32.
Dixit, A. K. (1980), “The Role of Investment in Entry-Deterrence”, Economic Journal, Vol. 90,
pp. 95-106.
Ohnishi, K. (2001), “Commitment and Strategic Firm Behaviour”, PhD Dissertation, Osaka
University.
Ohnishi, K. (2002), “On the Effectiveness of the Lifetime-Employment-Contract Policy”, The
Manchester School, Vol. 70, pp. 812-821.
Spence, M. A. (1977), “Entry, Capacity, Investment and Oligopolistic Pricing”, Bell Journal of
Economics, Vol. 8, pp. 534-544.
Ware, R. (1984), “Sunk Costs and Strategic Commitment: A Proposed Three-Stage Equilibrium”,
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