Schulz, Norbert

English

Research Papers in Economics

Nr. 99-5
COMMENT ON YIN, XIANGKANG AND YEW-KWANG NG:
QUANTITY PRECOMMITMENT AND BERTRAND COMPETITION YIELD
COURNOT OUTCOMES: A CASE WITH PRODUCT DIFFERENTIATION
Norbert Schulz
April 1999
University of Wuerzburg
Lehrstuhl für Volkswirtschaftslehre,
insbesondere Industrieökonomik
Sanderring 2, D-97070 Würzburg
norbert.schulz@mail.uni-wuerzburg.de
W. E. P.
Würzburg Economic Papers2
In a recent paper Yin, X. and Yew-Kwang Ng (1997) analyze a two stage game where
firms simultaneously set capacity first and then prices. In contrast to the celebrated paper
by Kreps and Scheinkman (1983) they consider a heterogeneous market. The authors argue
correctly that in this case there is no need to bother with the type of rationing procedure in
such a framework, if consumers are aware of the capacity choices. Therefore the critical
assessment of e.g. Davidson and Deneckere (1986) which stresses the importance of the
choice of rationing scheme are not relevant with equal force in their context. Their result
would support the original contribution of Kreps and Scheinkman in a more robust
situation. Unfortunately their analysis is not complete. More specifically their Lemma 1,
stating the continuity of the best response functions in the price game, is not correct. This
note concentrates on pointing out this defect. Some remedies are also proposed.
It turns out to be very helpful to start with a description of the demand behavior of
consumers as modeled by the authors. They assume a utility function of the form
ux x x x x x x xx x (,,) ( )( ) / 012 0 1 2 1
2
12 2
2 22 =+ + − + + αγ
where x0 denotes the quantity of some numeraire commodity and xi , i = 1, 2 denotes the
quantity of the consumed commodity supplied by firm i. Consumers are aware of the
capacities of the two firms Ki. Therefore their demand behavior is derived by maximizing
the utility function above under the following constraints:
Ix p x p x K x K x ≥+ + ≥≥ ≥≥ 01 12 2 11 22 00 ,,
The result of this optimization exercise can be conveniently summarized in the following
figure:
figure 1 should be inserted here
In the diamond shaped area we have interior solutions:
x
pp
1
1
2
2
2 1 11
=
+
−
−
+
−
α
γ γ
γ
γ
x
pp
2
2
2
1
2 1 11
=
+
−
−
+
−
α
γ γ
γ
γ
Let us turn next to the best response functions of firm 1. Consider first prices3
p2 ≤  α  - K2 - γ K1.
As long as p1 ≤  α  - K1 - γ K2 profits are increasing. Therefore p1 = α  - K1 - γ K2 will be the
global maximand within the rectangle containing the origin. Within the neighboring
trapezoid the profit maximizing choice is (α  - γ K2)/2, if this turns out to be larger than α  -
K1 - γ K2. If not, the maximizing price is α  - K1 - γ K2. Summarizing, the best response
function is given by
() pp K K K 12 1 2 22 ()m a x , ( ) / =− − − αγ α γ
for all prices p2 satisfying the restriction above. Note that
pp K K 12 1 2 () =− − αγ
for 2K1 + γ K2 ≤  α .
Next consider prices  p2 satisfying
α  - K2 - γ K1 ≤  p2 ≤  α  - K2.
For small prices p1- hence for pairs of prices in the left trapezoid - profits again are
increasing. Within this area the maximizing prices are therefore determined by the
boundary of this area:
pp K 12
2
1 11 =− + −− () ( ) γα γ γ
For higher prices such that the pair of prices are within the diamond shaped area the
maximizing price is
p
p
1
2 1
2
=
−+ () γα γ
,
if the pair of prices defined by this equation is contained in the diamond shaped area. Note
that this pair of prices would lie in the trapezoid to the left for all prices p2 satisfying the
restriction above, if (2 - γ
2) K1 + γ K2 ≤  α . Together with our result for low prices p2 this
implies that the best response function coincides with the boundary between the left
trapezoid and the diamond, if 2K1 + γ K2 ≤  α .
If we restrict our attention to such capacities it is easy to see that the best response function
coincides for all prices p2 with the right boundary of the area with x1 = K1. Of course, this
response function is continuous indeed. The problem arises with higher capacities.4
To illustrate this point, let us consider a case where 2K2 + γ K1 < α  such that the best
response function of firm 2 coincides with the upper boundary of the area with x2 = K2. At
the same time let (2 - γ
2) K1 + γ K2 < α  < 2K1 + γ K2. According to the arguments above the
best response function will be (α  - γ K2)/2 for p2 ≤  α  - K2 - γ K1. For α  - K2 - γ K1 ≤  p2 ≤
α  - K2 the situation is more complex. Recall that under this restriction the maximizing price
within the left trapezoid and the diamond together is determined by the common boundary
of these two areas. But for relatively small prices p2 (prices slightly above α  - K2 - γ K1)
there is another local maximum at (α  - γ K2)/2, which is strictly greater than the boundary
point. It may be helpful to summarize this situation in the following figure:
figure 2 should be inserted here
Under the parameter restrictions above the fat lines correspond to local maxima of firm 1's
profit function for all p2 ≤  α  - K2. Clearly for p2 above the upper dashed line the global
maximum coincides with the common boundary of the left trapezoid and the diamond, and
for p2 below the lower dashed line the global maximum is (α  - γ K2)/2. By continuity of the
profit function along the curves of local maxima it must be true that the best response
function jumps strictly between the two dashed lines.
This proves that the best response function is not continuous in this case contradicting
Lemma 1 of the authors' paper. Note that this lack of continuity will arise whenever α  <
2K1 + γ K2. Hence the problem arises under quite general circumstances.
In general, discontinuous best response functions do not necessarily harm the existence of a
Nash equilibrium. However, as noted above, under the parameter restriction considered
here the best response function of firm 2 is the upper boundary of the area with x2 = K2.
And this means that the two best response function do not intersect. Hence there does not
exist a pure strategy equilibrium in the price stage of the game.
What remedies might be proposed? The most straightforward suggestion is, of course, to
explicitly analyze equilibria in mixed strategies. This would probably leave the main result
of the authors intact. I probed only very partially into this matter. But it seems that the
mixed strategy equilibria will at least partly involve pure strategies where capacities are not
fully employed. Hence as in the Kreps/Scheinkman paper, mixed strategy equilibria are not
expected to be the outcome of the entire game when capacities are chosen at their
equilibrium levels. If this remedy is employed it will be accompanied by a quite tedious
analysis of a multitude of cases.5
A second suggestion is to circumvent these problems by setting an upper bound on
capacities like Ki < α /(2+γ ). This enforces continuous best response functions and leaves
the analysis of the authors intact. While such an assumption could be rationalized by
prohibitive costs of capacities above such a bound, it would be very ad hoc and thus
unsatisfactory.
A third alternative might be to drop the assumption that consumers are aware of the
capacities chosen by the firms. This reintroduces the problem of rationing.. If however
consumers face large transaction costs when visiting both firms then firms will abstain
from charging high prices in order to profit from capacity constraints of the competing
firms. In this case rationing is not profitable. If this approach is taken, the necessity of
considering mixed strategy equilibria in prices does not exist. For an analysis along these
lines see Schulz (1999): It is basically for this reason that the remedy of an analysis of
mixed strategy equilibria was not pursued in any depth in this note, although it is of some
interest to validate the hypothesis that the results of the authors can be amended in this
way.
REFERENCES:
Davidson, C. and R. Deneckere (1986) "Long-Run Competition in Capacity, Short-Run
Competition in Prices, and the Cournot Model" Rand Journal of Economics, vol 17, pp.
404-415
Kreps, D. and J. Scheinkman (1983) "Quantity Precommitment and Bertrand Competition
Yield Cournot Outcomes" Bell Journal of Economics, vol 14, pp. 326-337
Schulz, N. (1999) "Capacity Constrained Price Competition and Entry Deterrence in
Differentiated Product Markets" Würzburg Economic Papers, 99-7
Yin, Xiangkang and Yew-Kwang Ng (1997) "Quantity Precommitment and Bertrand
Competition Yield Cournot Outcomes: A Case with Product Differentiation" Australian
Economic Papers, vol 36, pp. 14-22

Long-Run Competition in Capacity, Short-Run Competition in Prices, and the Cournot Model&quot;

A comment on Yin, Xiangkang and Yew-kwang Ng: Quantity precommitment and Bertrand competition yield Cournot outcomes:a case with product differentiation

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A comment on Yin, Xiangkang and Yew-kwang Ng: Quantity precommitment and Bertrand competition yield Cournot outcomes:a case with product differentiation

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