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4Subgame Perfect Nash Equilibrium in a Quality—Price Competition Model with Vertical and Horizontal Differentiation
Denis Kuzyutin1, Elina Zhukova2, Maria Borovtsova2
1 International Banking Institute,
60, Nevsky prospect, St. Petersburg, 191023, Russia e-mail address: [email protected]
2 St. Petersburg State University,
Faculty of Applied Mathematics and Control Processes,
35, Universitetskii prospekt, St. Petersburg, 198504, Russia e-mail address: [email protected]
Abstract. The paper describes how to construct the subgame perfect equilibrium by using a backwards induction procedure in a model of product differentiation that takes into account both vertical and horizontal differentiation features.
Keywords: Industrial organization, product differentiation, subgame perfect Nash
equilibrium.
1. Presuppositions and Formalization of the Game-Theoretical Model with Vertical and Horizontal Differentiation Features
We propose a game-theoretical duopoly model of product differentiation that takes into account both vertical and horizontal differentiation features. The research is based on the following assumptions:
• Each consumer estimates the quality of the product and this estimation is important for determining the upper level of the price acceptable for the regarded consumer.
• Each consumer is characterized by a parameter t : t G [t, i] C [0, +oo) which shows his willingness to pay for quality increasing, and all consumers range the substitute goods in the same way when their prices are equal.
• If a firm wants its product to be higher estimated by consumers it has to invest in R&D and generally it leads to the growth of both fixed and variable costs.
• Two competitive firms are supposed to be located on the different sides of a “linear city” ([Hotelling, 1929]; [Tirole, 1997]). All consumers are uniformly distributed on the segment between these points.
• Travel costs of the consumer are in direct proportion to the distance between the consumer and the firm.
• Firms play a two-staged game. In the first stage they choose simultaneously their respective quality level. In the second stage they choose simultaneously their price. Quality decisions are observable by both rivals before price decisions are made.
Now, let’s regard the game more properly.
In the 1st step (at the stage of their quality decisions) both firms choose their qualities simultaneously: </j G [q,q] and, thus, they face some quality development costs FC(qi). For each one of the consumers (we suppose S consumers in all) the products of the firms are substitute goods. Each consumer has to choose between purchasing one unit of product from one of the firms or making no purchase.
The 1st stage decisions (qi, q2), q1 < q2 become observable before the second stage starts. At the second stage (the stage of price-competition) the rivals choose their prices p1 and p2 respectively. Here and further the subscript 2 denotes the firm that has entered the market with a quality that exceeds the quality of the other firm. This firm is labeled “the high-quality seller”. The other firm is labeled “the low-quality seller” and is denoted by the subscript 1.
Each consumer’s strategy is maximizing his utility function of the following form:
Ut = max{tq2 — p2 — k(p - s), tqi - pi - k(s - p), 0}, (1)
where t : t G [t, t\ C [0, +oo) - is a scalar parameter which shows consumer’s willingness to pay for quality increasing; s : s G [p, ~p\ C [0, +oo) - is a scalar parameter that characterizes consumer’s spatial position on the segment [p, p]; k - is a scalar parameter that can be interpreted as transport costs. Parameters t and s together can be regarded as a 2-dimensional variate uniformly distributed on a rectangle [t, t] x \p, ~p\.
Consumer’s reaction described above leads to self-segmentation of the consumers and unambiguously determines market shares D1 and D2. The consumer t will buy the hight quality product if and only if his surplus from buying the high-quality product will be higher than from buying the low-quality product or
tq2 -P2~ k(p - s) > tqi -p\- k(s - p).
And the position of marginal consumer t2(s) who is indifferent between purchasing low quality or high quality product can be defined by solving:
. ,, ^ Hp +p-2s)-Pl+P2
tq2 — P2 — k{p — s) = tqi — p\ — k{s — p) => t2{s) = -------------------------. (2)
- q2-qi
The area above the line t2(s) is the high-quality seller’s market share: D2. Similarly
Fig.1 : Self-segmentation of the consumers
we determine the low-quality seller’s market share: Di. In this case the position of marginal consumer who is indifferent between buying the low-quality product or making no purchase is a line t1(s). Its equation can be analytically found by solving:
pi + k(s — p)
tqi - pi - k(s - p) = 0 => ti(s) = ---------------(3)
_ Q l
The consumers under the line t1(s) make no purchase. The market shares D1 and D2 are proportional to the areas of the corresponding figures (see Figure 1). The purpose of each firm is maximizing its profit. The solution concept traditionally used in similar models [Gabszewics, 1979], [Sutton, 1982], [Ronnen, 1991], [Motta, 1993], [Petrosjan, 1983], [Zenkevich, 2006] is a subgame perfect equilibrium SPE [Selten, 1975]. The equilibrium is solved by the backwards induction method. Here and further we suppose nontrivial situation when q2 > q1.
2. The Construction of Second-stage (Price) Equilibrium
Now let us consider some private case of the model with elements of vertical and horizontal differentiation described above:
o q G (0, +oo); [t,t] = [0, 1]; S = 1.
o FC(q) = ^q2.
o t2(p) < l]t2(p) > hip). (4)
Denote d = ~p — p. The last conditions in the list restricts the research by the case
when the lines ti(s) and t2(s) do not intersect within the area regarded and t2(p) < 1
(like in Figure 1).
According to the backwards induction procedure we begin from the second (and the last) stage of the game - the stage of price competition.
Let q1 and q2 be the first stage firm’s quality decisions.
Theorem 1. In the game-theoretical model of duopoly with vertical and horizontal differentiation features there exists second-stage price equilibrium:
[ P\^m) = {q^Xkd\
{ (5)
1 p£(gi,g 2) = {q27{lli%kd) ■
Proof.
In order to find the second-stage price equilibrium we construct the reaction functions of the firms.
Firstly, we construct the reaction function of the high-quality seller. Notice that a consumer (t, s) will buy a product of this firm if the point (t, s) is located above the curve t2(s)(see Figure 1). Thus, we can conclude that the high quality seller’s
market share is
D2 =
d(q2 - ql - P2 + Pl)
q2 - q1
And the high-quality seller’s profit will be:
tt / ^ n «2 d(q2 - qi - P2 + Pi) q2
U-2{qi,q2,Pl,P2) =P2P>2 - -77 = P2--------------------------------^ ■
2 q2 — q1 2
Thus, the reaction function of the high quality seller will look like
P2(Pl) =
where p1 meets the following conditions (according to the conditions (4)):
P1 <
q2 — ql — 2kdr
2r - 1 Pl > 2kd — q2 + ql.
The low-quality seller’s market share is
D1 =
2d(p2qi ~Piq2) ~ kd2(q2 - qi) Zqi(q2 - qi) '
(6)
(7)
(8)
(9)
(10)
(11)
Its profit will be
TT , N n Qi 2d(p2qi - Piq2) ~ k(P(q2 - qi) q\
Hi{qi,q2,pi,p2) =P\Di - — =Pl-----------------—----------------------------- -—. 12
2 2q1(q2 — q1) 2
And the reaction function of the low-quality seller will be
2q1P2 — kd(q2 — q1)
P2{Pl) = ------------------------, 13
4q2
where p2 meets the following conditions (according to the conditions (4)):
k > + (14) 2(2q2 — q1r)
4q2(q2 — q1 — kd) — kd(q2 — q1)
K < --------------2^1------------------ (15)
The two reaction curves intersect uniquely and give the vector of equilibrium prices (5).
3. The Equilibrium at the Stage of Quality Competition
Let us note that when prices p1 and p2 are selected the profits of the rivals are equal to
tt / ^ Q2d(q2 - qi)(qi - kd)2 1 2
ni(,1'®) =--------„(4,2-91)2-----------2*' ' ’
tt / \ d(q2 - qi)(4:q2 ~ kd)2 1 2
=-------4(4® -qif------------2^ (17)
In accordance to the subgame perfect equilibrium concept now we regard the first stage of the game - the stage of quality competition. If the vector of selected qualities (q*, q*) is the equilibrium, the following conditions should be held:
(18)
ff(^2*)=0.
The system (18) have been numerically solved for the private case: p = 1, p = 0, /c = 0,01.
The following decisions have been found:
A. q*A & 0, 0063, q*A & 0, 0041;
B. q*B & 0, 0009, q*B & 0, 0025;
C. q*C & 0, 0111, q*C & 0, 2502;
D. q*D & 0, 0485, q*D & 0, 2528.
We had supposed q1 < q2, thus, the decision A mismatches. The decisions B and C do not match either because n1(q* C, q*C) < 0 and n2(q*B, q*B) < 0.
Now let’s check whether the vector (q*D,q*D) meeting the conditions (18) is a
Nash equilibrium.
Let the firm 1 deviates from q* D and chooses some q 1 < q*D. In this case the profit of the low-quality firm will be:
nihi.gSc) = °’2528(0’;528-«)(« -°’01)2 - i. (19)
</i(l, 0112 — </i)2 2 V ’
Investigating the derivative §^(</1, q2]j), we conclude that the low-quality seller’s profit is a decreasing function in the segment (-0,009; 0,0115) and (0,048;1,7501). Considering conditions: 0, 01 < q1 < q*D, and
^(0,01,^) = -^ <0, (20)
we receive
arg max n1(q1,q*D) = q*D & 0,0485. (21)
0,01<qi<q2D
We have, therefore, proved that the low-quality seller (firm 1) has no incentive to deviate from the strategy q1 D .
In a similar manner it is possible to prove that firm 2 won’t deviate from its strategy q*D on the segment (q*D, +ro) either.
Besides, it is also necessary to check that the 1st firm will not deviate from q*D on the segment (q*D, +ro), i.e. that the 1st firm has no incentive to “jump over” the quality q2 D and to become the high-quality seller.
Let the 1st firm choose some quality q1 > q*D. In this case it’s profit will be
n,fai,q3c) = -°’2528)(4« - °'01)i _ gj. ,,2)
4(49i- 0,2528)2 2 V ’
Investigating the derivative Q2d) we conclude that the 1st firm’s profit is a
decreasing function in the segment (0, 3337; +ro). Considering q1 > q*D we receive:
arg max n1(q1,q* D) & 0,3337. (23)
q£D <qi<+TO
The value of profit n1(0, 3337, q*D) = —0,0253 < 0, i.e. it isn’t advantageous for the 1st firm to deviate from the value q* D in the segment (q*D, +ro).
In a similar manner it is possible to prove that firm 2 won’t “jump over” the quality q1 d and become the low-quality seller. Therefore, the pair of the rival’s strategies
(q* d,P*(q*d,q*d)) and (q*d,P2(q*D,q*D)) make a subgame perfect equilibrium in the game. In this equilibrium qualities, prices and profits are equal accordingly
q*D & 0, 0485, q*D & 0, 2528;
p1 & 0, 0082, p2 & 0,1062; (24)
n & 0, 0005, n2 & 0, 0233.
Thus, we managed to construct the subgame perfect equilibrium in a proposed 2-space (vertical and horizontal) product differentiation model using the backwards induction procedure.
References
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