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QUANTITY PRECOMMITMENT AND BERTRAND COMPETITION
YIELD COURNOT OUTCOMES: A CASE WITH PRODUCT
DIFFERENTIATION: REPLY
XIANGKANG YIN
La Trobe University
YEW-KWANG NG
Monash University
Since the publication of our paper (Yin and Ng, 1997), we have received some very
helpful comments on the paper. In his recent comment, Schulz (2000) indicates that price
reaction functions in the second-stage subgame of our model are discontinuous in certain
circumstances so that Lemma 1 in the paper is incorrect. The editor, R. Damania, points out
a mistake in the proof of Lemma 1 in his letter to us. We sincerely appreciate these
criticisms, which are greatly helpful for us to re®ne our work. After a careful check, we ®nd
that not only Lemma 1 is invalid but the price reaction function given in Theorem 1 also has
to be revised. Fortunately, these errors do not affect the main results of the paper. The
erratum to the previous mistakes is given below.
Figure 1 below is borrowed from Schulz (2000), which can be obtained by applying
demand functions (3)±(7) in our original paper. In the ®gure, line Ci (i ˆ 1, 2) plots the price
pairs on which the demand for the ®rm i's product just hits its capacity while the demand for
the ®rm j's product is not binding by either the upper or lower boundary; i.e.
Ci : p1i ( p j ) ˆ (1 ÿ ã)á ‡ ã p j ÿ (1 ÿ ã2 )K i :
(1)
Z i in the ®gure (i ˆ 1, 2) is the line on which the demand for the ®rm i's product just
becomes zero while the demand for the ®rm j's product hits neither the upper nor lower
boundary; i.e.
Z i : pi ˆ (1 ÿ ã)á ‡ ã p j : Or inversed, p2j ( pi ) ˆ [ pi ÿ (1 ÿ ã)á]=ã:
(2)
Axy gives the area, where the demand for ®rms 1 and 2's products has the characteristics of
the ®rst and second subscripts, respectively. x and y in the subscript can be c, u, and z,
indicating that the demand is constrained by the production capacity, unconstrained demand
and zero demand, respectively. The equations below Axy show the demand functions in that
area and the demand in Auu is q i ˆ á=(1 ‡ ã) ÿ ( pi ÿ ã p j )=(1 ÿ ã2 ).
It should be noticed that Figure 1 presents a most comprehensive situation with nine areas,
which requires small production capacities of both ®rms. When capacities are large, C1 and
C2 move to left and down, respectively, so that some areas may disappear. An extreme is no
capacity constraints and Figure 1 has only four areas: Auu, Auz, Azu, and Azz. With the
help of Figure 1, we can revise the price reaction function as follows.
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Figure 1.
Theorem 19 (I) If 2Ki ‡ ãK j < á, the price reaction function is1
8
á ÿ K i ÿ ãK j
p j 2 [0, á ÿ K j ÿ ãK i ]
>
>
<
1
p j 2 (á ÿ K j ÿ ãK i , á ÿ ãK i )
Ri ( p j ) ˆ p i ( p j )
>
>
:
2
minf p i ( p j ), á=2g p j 2 [á ÿ ãK i , á]
(II) If 2Ki ‡ ãK j . á, the price reaction function is
8
(á ÿ ãK j =2
p j 2 [0, á ÿ K j ÿ ãK i ]
>
>
>
>
>
>
p j 2 (á ÿ K j ÿ ãK i , á ÿ ãK i )\
(á ÿ ãK j )=2
>
>
>
>
>
>
>
^p1 g)
(á ÿ K j ÿ ãK i , minf~p1 , ^
>
>
<
1
3
p j 2 (á ÿ K j ÿ ãK i , á ÿ ãK i )\
Ri ( p j ) ˆ maxf p i ( p j ), p i ( p j )g
>
>
>
>
^p1 g, á ÿ ãK i , )
(minf~p1 , ^
>
>
>
>
>
>
>
^p1 g)
p j 2 [á ÿ ãK i , á] \ (á ÿ ãK i , minf p1 , ^
(á ÿ ãK j )=2
>
>
>
>
:
^p1 g, á]
minfmax[ p2 ( p j ), p3 ( p j )], á=2g p j 2 [á ÿ ãK i , á] \ (minf p1 , ^
i
i
1
An interval (x, y) below should be understood as an empty set if x > y or both x and y are negative.
If x , 0 but y . 0, the interval (x, y) should be understood as [0, y).
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115
Proof. (I) Consider 2K 2 ‡ ãK 1 < á.
p1 2 [0, á ÿ K 1 ÿ ãK 2 ]
(i)
For a given p1 , ®rm 2 chooses the optimal price p2 ˆ á ÿ K 2 ÿ ãK1 if q2 ˆ K 2 or p2 ˆ
(á ÿ ãK 1 )=2 if q2 ˆ á ÿ p2 ‡ ãK 1 . The latter is the best price response iff it is in A cu .
Since 2K 2 ‡ ãK 1 < á , á ÿ K 2 ÿ ãK1 > (á ÿ ãK 1 )=2, the price reaction function is
á ÿ K 2 ÿ ãK1 .
p1 2 (á ÿ K 1 ÿ ãK 2 , á ÿ ãK 2 )
(ii)
There are two candidates of optimal price. One is to choose maximum price when the
capacity is binding, which is the line C2 , i.e., p12 ( p1 ). The other is
p32 ( p1 ) ˆ [(1 ÿ ã)á ‡ ã p1 ]=2,
(3)
2
which maximises pro®ts for demand q2 ˆ á=(1 ‡ ã) ÿ ( p2 ÿ ã p1 )=(1 ÿ ã ). Recalling (1),
p1 . á ÿ K 1 ÿ ãK 2 and 2K 2 ‡ ãK1 < á, we have
p12 ( p1 ) ˆ [(1 ÿ ã)á ‡ ã p1 ]=2 ‡ [(1 ÿ ã)á ‡ ã p1 ÿ 2(1 ÿ ã2 )K 2 ]=2
. p32 ( p1 ) ‡ [á ÿ ãK 1 ÿ 2K 2 ‡ ã2 K2 ]=2 . p32 ( p1 ):
Since p32 ( p1 ) is the best price response iff it falls in A uu , the above inequality implies that
the price reaction function in this interval must be p12 ( p1 ).
p1 2 [á ÿ ãK 2 , á]
(iii)
(I)±(ii) shows that p32 ( p1 ) , p12 ( p1 ) when p1 2 (á ÿ K 1 ÿ ãK 2 , á ÿ ãK 2 ). Thus, p32 ( p1 ) ,
á ÿ K 2 . Since p32 ( p1 ) is ¯atter than Z 1 , it is in A zc or A zu on this interval and cannot be the
best price response. Noting that p2 ˆ á=2 maximises pro®t for demand q2 ˆ á ÿ p2, the
best price response should be Z 1 if p2 ˆ á=2 is above Z 1 or p2 ˆ á=2 if it is below Z 1 ; i.e.
minf p22 ( p1 ), á=2g.
(II) Assuming 2K 2 ‡ ãK 1 . á
(i)
p1 2 [0, á ÿ K 1 ÿ ãK 2 ]
As (I)±(i) shows the price reaction function on this interval is (á ÿ ãK 1 )=2.
(ii)
p1 2 (á ÿ K 1 ÿ ãK 2 , á ÿ ãK 2 )
In this interval, ®rm 2 has three possible price choices, i.e., p2 ˆ (á ÿ ãK1 )=2 or p12 ( p1 ) or
p32 ( p1 ). Their corresponding pro®ts are, respectively
(á ÿ ãK 1 )2 =4,
K 2 p12 ( p1 ) ˆ K 2 [(1 ÿ ã)á ‡ ã p1 ÿ (1 ÿ ã2 )K 2 ],
fá=(1 ‡ ã) ÿ [ p32 ( p1 ) ÿ ã p1 ]=(1 ÿ ã2 )g p32 ( p1 )
ˆ [(1 ÿ ã)á ‡ ã p1 ]2 =4(1 ÿ ã2 )
Let p~1 be the point where the ®rst pro®t ®gure is equal to the second, we have
~p1 ˆ [(á ÿ ãK 1 )2 =4K 2 ‡ (1 ÿ ã2 )K 2 ÿ (1 ÿ ã)á]=ã:
(4)
Thus, p2 ˆ (á ÿ ãK 1 )=2 leads to a larger pro®t than p12 ( p1 ) iff p1 , ~p1 . Let p^1 be the point
where the ®rst pro®t ®gure is equal to the third, we have
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^p1 ˆ [(1 ÿ ã2 )1=2 (á ÿ ãK 1 ) ÿ (1 ÿ ã)á]=ã,
(5)
iff p1 , ^p1 . Let p1 be
which implies that p2 ˆ (á ÿ ãK 1 )=2 brings more pro®t than
the intersection of p12 ( p1 ) and p32 ( p1 ); i.e., p1 ˆ [2(1 ÿ ã2 )K1 ÿ (1 ÿ ã)á]=ã. Since p12 ( p1 )
is steeper than p32 ( p1 ), we can see that p32 ( p1 ) . p12 ( p1 ) on the left of p1 and
p12 ( p1 ) > p32 ( p1 ) on the right. Moreover, for any p1, if p32 ( p1 ) falls in A uu , it results in more
pro®t than p12 ( p1 ). It can also be shown that ~p1 . á ÿ K 1 ÿ ãK 2 and p32 (^p1 ) is below the
line C1 (or its extension). Thus, the price reaction function shifts from p2 ˆ (á ÿ ãK1 )=2 to
p32 ( p1 ) at p1 ˆ ^p1 if ^p1 , ~p1 and ^p1 2 (á ÿ K 1 ÿ ãK 2 , p2 ),2 and then switches to p12 ( p1 ) at
= (áÿ K1 ÿ ãK 2 , p2 ),
p1 ˆ p1 if p1 , á ÿ ãK 2 . But if ~p1 < ^p1 or ~p1 < á ÿ ãK 2 and ^p1 2
1
the price reaction function shifts from p2 ˆ (á ÿ ãK 1 )=2 to p2 ( p1 ) at p1 ˆ ~p1 and remains
on it until the end of the interval. If ~p1 . á ÿ ãK 2 and ^p1 . á ÿ ãK 2 or ~p1 . á ÿ ãK 2 and
^p1 2
= (á ÿ K 1 ÿ ãK2 , p2 ), the reaction function remains on p2 ˆ (á ÿ ãK 1 )=2 for the
whole interval. De®ne
(
^p1 ^p1 2 (á ÿ K 1 ÿ ãK 2 , p2 )
^^p1 ˆ
:
(6)
1 otherwise
p32 ( p1 )
The price reaction function in this interval is
(á ÿ ãK 1 )=2
^p1 g
when p1 < minf~p1 , ^
^p1 g
maxf p22 ( p1 ), p32 ( p1 )g when p1 > minf~p1 , ^
:
p1 2 [á ÿ ãK 2 , á]
2
Let p1 be determined by p2 ( p1 ) ˆ á=2 so that p1 ˆ (1 ÿ ã=2)á. As (I)±(iii) shows, the
price reaction function cannot be p2 ˆ á=2 on the left of p1 . Hence, on the left of p1
there are three possible price choices, i.e., p2 ˆ (á ÿ ãK 1 )=2 or p22 ( p1 ) (i.e., line Z 1 ) or
p32 ( p1 ). The pro®t of adopting p22 ( p1 ) is equal to fá ÿ [ p1 ÿ (1 ÿ ã)á]=ãg[ p1 ÿ
(1 ÿ ã)á]=㠈 (á ÿ p1 )[ p1 ÿ (1 ÿ ã)á]=ã2 . Let p1 be the point where this pro®t is equal to
(á ÿ ãK 1 )2 =4, i.e., p1 is the smaller solution of the equation
(iii)
p21 ÿ (2 ÿ ã)á p1 ‡ (1 ÿ ã)á2 ‡ ã2 (á ÿ ãK 1 ]=4 ˆ 0:
(7)
2
and
is on the left of p1 and p2 ( p1 ) is
It can be shown that the intersection of
steeper than p32 ( p1 ). Then, if the price reaction function is p2 ˆ (á ÿ ãK1 )=2 at
^p1 , p1 . Afterwards, it switches to
p1 ˆ á ÿ ãK 2 , it shifts to p32 ( p1 ) at the p1 ˆ ^^p1 when ^
^p1 it shifts to
p22 ( p1 ) at the intersection of p22 ( p1 ) and p32 ( p1 ) to reach p2 ˆ á=2. When p1 , ^
p22 ( p1 ) at p1 ˆ p1 and moves toward á=2. If the price reaction function at p1 ˆ á ÿ ãK 2 is
p22 ( p1 ) or p32 ( p1 ), the development is similar, except that there is no shift. Thus, the price
reaction function in this interval is
p22 ( p1 )
(á ÿ ãK 1 )=2
p32 ( p1 )
^p1 g
when p1 < minf p1 , ^
^p1 g:
minfmax[ p22 ( p1 ), p32 ( p1 )], á=2g when p1 > minf p1 , ^
j
With this new Theorem 19, Lemma 1 in our original paper should be changed to
2
The second condition is to ensure that p32 (^p1 ) falls in A uu .
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Lemma 19. Firm i's price reaction function has one and only one discontinuity point at
^p j when 2K i ‡ ãK j . á and ãK j , á. Otherwise, it is conninuous.
p j ˆ p j or ~p j or ^
Proof. Since maxf:, :g and minf:, :g are continuous operators, the discontinuity occurs at
the point where the price reaction function shifts from pi ˆ (á ÿ ãK j )=2 to p1i ( p j ) or p2i ( p j )
or p3i ( p j ). But this can happens only once when 2K i ‡ ãK j . á and ãK j , á as Theorem 19
illustrated.
j
The bold lines in Figure 2 illustrates a typical price reaction curve of ®rm 2 when the
^p1 . If the
discontinuity conditions in Lemma 19 are satis®ed. Thus, there is a shift at p1 ˆ ^
line p32 ( p1 ) is lower, the shift may occur at p1 ˆ ~p1 . p1 . A rare situation is that K 1 is very
small but K 2 is suf®ciently large so that that the line C1 is long but C2 is short. In turn, the
shift occurs on the right of p1 ˆ á ÿ ãK 2 .
Because of the discontinuity of the price reaction curves, the price subgame in the second
stage does not guarantee a unique pure strategy equilibrium as indicated in Theorem 2 in the
original paper. However, this theorem is not a necessity for Theorem 3. Hence, Theorem 3
holds for the discontinuous price reaction functions but the proof should be revised.
Theorem 3. In the equilibrium of the entire game, there are no idle capacities.
Proof. Suppose ( p1e , p2e , K 1 , K 2 ) is the equilibrium of the entire game and at least one
®rm, say, ®rm 2, has certain idle production capacity. If the equilibrium leads to a pure price
Figure 2.
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strategy of ®rm 2, then ( p1e , p2e ) must be on the line p2 ˆ (á ÿ ãK 1 )=2 or p22 ( p1 ) or p32 ( p1 )
or p2 ˆ á=2. Hence, p2e is independent of ®rm 2's capacity and it has an incentive to reduce
its idle capacity, which implies that ( p1e , p2e , K 1 , K 2 ) cannot be the equilibrium of the entire
game.
Consider now that ®rm 2 adopts a mixed price strategy in equilibrium. Since the mixed
price strategy can only occur at the discontinuity point of the price reaction function, it
^p1 or p1 or ~p1. For the ®rst two cases, ( p1e , p2e ) is the intersection of the
implies that p1e ˆ ^
^
line C1 and p1 ˆ ^p1 or C1 and p1 ˆ p1 . Because C1 , ^^p1 and p1 are independent of K 2 (see
(1), (5)±(7)), the price p2e is independent of ®rm 2's capacity.
Turning to the case where p1e ˆ ~p1 . Figure 3 illustrates the most complicated situation in
the sense that both ®rms have discontinuous price reaction functions, resulting in multiequilibria at E1 and E2 in the price subgame.3 Let us focus on E1 . The shift of price reaction
curve at ~p1 from p2 ˆ (á ÿ ãK 1 )=2 to p12 ( p1 ) (i.e., line C2 ) implies that p12 (~p1 ) . p32 (~p1 ).4
Recalling (1), (3) and (4), this inequality gives (1 ÿ ã)á ‡ ã~p1 . 2(1 ÿ ã2 )K 2 and
(á ÿ ãK 1 )2 =4K 2 . (1 ÿ ã2 )K 2 :
(8)
Since E1 is on C1 , we obtain from the reverse of p11 ( p2 ) that
p2e ˆ [~p1 ‡ (1 ÿ ã2 )K 1 ÿ (1 ÿ ã)á]=ã:
(9)
The demand for good 2 along line C1 is q2e ˆ á ÿ p2e ÿ ãK 1 . Thus, the ®rm 2's equilibrium
pro®t in the second-stage subgame is ð ˆ p2e q2e ˆ p2e (á ÿ p2e ÿ ãK 1 ), which gives
@ð=@ K 2 ˆ (á ÿ 2 p2e ÿ ãK 1 )@ p2e =@ K 2
ˆ ãÿ2 (á ÿ 2 p2e ÿ ãK1 )[(1 ÿ ã2 ) ÿ (á ÿ ãK 1 )2 =4K 22 ] , 0:
Figure 3.
3
In equilibrium E2 , the ®rm 2's capacity is binding. But actually, we can show that it is not the
equilibrium of the entire game following the same arguments in the proof of E1 . If ®rm 1's price
reaction function is continuous, then there exists only one equilibrium, E1 . The intersection of ~p1 and
~p2 cannot be an equilibrium point because it falls in the area A uu .
4
Otherwise, it shifts to p32 ( p1 ) before ~p1 .
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To get the second equality, (4) and (9) are used. From (8) and q2e . á ÿ 2 p2e ÿ ãK 1, the
inequality is immediate, which implies that ®rm 2 has an incentive to reduce the capacity
j
and ( p1e , p2e , K1 , K 2 ) is not an equilibrium.
When the capacity constraints of both ®rms are binding, the price reaction function is
Ri ( p j ) ˆ á ÿ K i ÿ ãK j , which leads to the price subgame equilibrium (11) in our original
paper. Therefore, the main conclusions, Theorem 4 and Proposition 1 there, hold.
RE F ERE N C E S
Schulz, N. 2000, `A Comment on Yin, Xiangkang and Yew-Kwang Ng: Quantity Precommitment
and Bertrand Competition Yield Cournot Outcomes: A Case with Product Differentiation'
Australian Economic Papers, vol. 39, pp. 108±112.
Yin, X. and Ng, Y.-K. 1997, `Quantity Precommitment and Bertrand Competition Yield Cournot
Outcomes: A Case with Product Differentiation' Australian Economic Papers, vol. 36, pp. 14±
22.
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