Endogenous choice of price or quantity contract
with upstream advertising
Qing Hu
Dan LiTomomichi Mizuno
January 2023
Discussion Paper No. 2301

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANEndogenous choice of price or quantity contract with
upstream advertising∗
Qing Hu†ac , Dan Li‡b , Tomomichi Mizuno§ca Faculty of Economics, Kushiro Public University of Economicsb School of Management, Xi’an Polytechnic University
c Graduate School of Economics, Kobe University

January 17, 2023
Abstract
We investigate a supply chain comprising a manufacturer engaged in advertising andtwo retailers who compete with diﬀerentiated products. We examine the endogenous
choice between competing on quantity or price for the retailers. Our analysis reveals
that, depending on the level of product substitutability, the range of possible outcomesis varied and includes Cournot, Bertrand, and Cournot-Bertrand under informative
advertising. This result contradicts the established understanding that ﬁrms tend toengage in Cournot competition as their dominant strategy. Furthermore, we ﬁnd that
under persuasive advertising, Cournot or Bertrand outcomes may be optimal, but
Cournot-Bertrand never arises as an equilibrium.

JEL codes: D43, L13, M21.
Keywords: endogenous competition mode, advertising, vertical relationship.
∗

The third author gratefully acknowledges the ﬁnancial support from the Japan Society for the Promotionof Science, KAKENHI Grant Numbers JP22H00043 and JP20K01678. The usual disclaimer applies.
†
Corresponding author: Qing Hu, Faculty of Economics, Kushiro Public University of Economics, Ashino
4-1-1, Kushiro-shi, Hokkaido, 085-8585, Japan. Phone: (81)-154-37-3211. E-mail: huqing549@gmail.com‡
Dan Li, School of Management, Xi’an Polytechnic University, 19 JinHua NanLu, Xi’an, 710048, China.
E-mail: catherinedanli@hotmail.com§
Tomomichi Mizuno, Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, Hyogo
657-8501, Japan. E-mail: mizuno@econ.kobe-u.ac.jp1

1

Introduction

We analyze the endogenous choice between quantity or price competition faced by retailers
in a vertical market with an upstream manufacturer engaged in advertising and two downstream retailers competing with diﬀerentiated products. Singh and Vives (1984) analyzed
the endogenous choice in the absence of advertising by the upstream manufacturer, using
Cournot, Bertrand, and Cournot-Bertrand static duopoly models within the framework of
Dixit (1979). They demonstrated that when products are substitutes, Cournot competitionis the dominant strategy for ﬁrms. In this study, we challenge this conventional wisdom byconsidering the inﬂuence of advertising by the upstream manufacturer.
We consider a four-stage game played by a manufacturer and two retailers oﬀering differentiated products.1 In the ﬁrst (pre-play) stage, the retailers simultaneously choose thetype of market contract (quantity or price). In the second stage, the manufacturer chooses
the level of advertising, which aﬀects the size of the retailers’ markets. In the third stage,
the manufacturer decides the wholesale price. In the fourth stage, the retailers compete inaccordance with the market contract choices made in the ﬁrst stage.
We ﬁnd that in the case of informative advertising, retailers’ market contract choices
depend on the degree of product substitutability. Speciﬁcally, when the degree of product substitutability is suﬃciently high, both retailers choose to oﬀer a quantity contract(Cournot) in equilibrium, consistent with the result in Singh and Vives (1984). However,when the product substitutability is suﬃciently low, both retailers may choose to oﬀer aprice contract (Bertrand), in contrast with the result in Singh and Vives (1984). Additionally, when the product substitutability is at an intermediate level, there may be two
asymmetric subgame perfect Nash equilibria (SPNE) in the ﬁrst stage, with one retailer
oﬀering a price contract and the other oﬀering a quantity contract (Cournot-Bertrand). The
retailers’ choices are strategic substitutes, which never occurs in the model provided by Singhand Vives (1984).

1

The literature on advertising (Gross and Shapiro, 1984; Soberman, 2004; Zhang et al., 2012; Zhang etal., 2020) allows ﬁrms to sell diﬀerentiated products.

2

The intuition underlying this result is as follows: It is well-known that the market is themost competitive under Bertrand competition, relatively less competitive under the CournotBertrand market, and the least competitive under Cournot competition. The manufacturer
tends to advertise more in a more competitive retail market to increase output, which beneﬁts
retailers by increasing demand. Simultaneously, a more competitive market adversely aﬀects
retailers’ proﬁts. When the degree of product substitutability is low, the scale of advertising
is also small. In this case, the retailer has an incentive to intensify competition to encourage
advertising investment. If product substitutability is suﬃciently low, both retailers choosea price contract. As the degree of product substitutability increases, the incentive to selecta price contract decreases. At some level of product substitutability, if one retailer choosesa price contract, the other retailer loses the incentive to choose a price contract because
the manufacturer’s advertising investment increases discontinuously. Consequently, for anintermediate level of product substitutability, the retailers choose asymmetric contracts.
Finally, in the case of high product substitutability, no retailer chooses a price contract
because the level of manufacturer advertising is suﬃcient.
Additionally, we ﬁnd that in the case of persuasive advertising, a Bertrand outcome
occurs in equilibrium when the advertising investment is suﬃciently eﬃcient or the degreeof product substitutability is suﬃciently low. When the degree of product substitutability
is low, both retailers may choose a price contract, which is similar to the case of informative
advertising. Additionally, when advertising investment is eﬃcient, the beneﬁcial advertising
eﬀect is suﬃciently strong to persuade one retailer to choose a quantity contract, leading
to a Cournot outcome. However, when advertising investment is ineﬃcient or the degreeof product substitutability is intermediate, a Cournot-Bertrand outcome never occurs inequilibrium. Instead, either a Cournot or a Bertrand outcome may be obtained.
Our study is related to previous research that has examined the well-known Cournot
advantage property of endogenous choice between quantity or price competition in various
contexts (Zanchettin, 2006; Arya et al., 2008; Tremblay et al., 2009; Basak and Wang,2016). These studies have demonstrated that the subgame perfect equilibrium could be3

Bertrand or Cournot-Bertrand. However, all of these studies considered models without
upstream manufacturer advertising. Our work is closely related to that of Hu and Mizuno(2021), who investigated retailers’ second-mover advantage property of Bertrand competition
by introducing advertising by the upstream manufacturer. They found that simultaneous
pricing may occur in equilibrium when product substitutability is suﬃciently low. Our
study complements their work by analyzing the endogenous choice between quantity or pricecompetition faced by retailers by considering Cournot, Bertrand, and Cournot-Bertrand in
the presence of both informative and persuasive advertising by the upstream manufacturer.
In Hu and Mizuno (2021), the eﬃciency of advertising by the manufacturer does not play asigniﬁcant role; however, in our model, advertising eﬃciency plays an important role in thedetermination of retailers’ market contract choices under persuasive advertising.
Additionally, we ﬁnd that the assumption that the retailers choose the type of contract
before informative advertising is crucial for our main results. Furthermore, we demonstratethat the result of Singh and Vives (1984) still holds in a vertical structure with an upstream
manufacturer, but it does not hold when informative advertising by the manufacturer isintroduced. This highlights the importance of advertising by the manufacturer for retailers’market contract decisions.
The remainder of the paper is organized as follows: In Section 2, we describe the basicmodel. In Section 3, we analyze retailers’ market contract choices in the case of informative
advertising. In Section 4, we present the game in reversed timing and analyze the case of
persuasive advertising. Finally, in Section 5, we present our conclusion.

2

Model

We consider a market with a manufacturer and two retailers, referred to as retailer i andretailer j (i, j = 1, 2 and i ̸= j). The retailers purchase products at a wholesale price w
and sell it to consumers. The products sold by the retailers are diﬀerentiated (e.g. eachretailer sells a product with a diﬀerentiated service). The retailers choose either their priceor output, and the price and output of retailer i are denoted by pi and Qi , respectively. The
4

proﬁt of retailer i is then given by πi ≡ (pi − w)Qi .
The manufacturer can engage in informative advertising. We denote the level of informative advertising by θ, which represents the fraction of consumers who receive the advertisement. Consumers who view the advertisement can buy either product, while those who do
not view it buy none. We assume that if θ consumers receive the advertisement, the manufacturer must incur an advertising cost of kθ2 , where k is a positive and constant parameterand θ ∈ [0, 1].2 The manufacturer produces the product at a constant marginal cost of c.
Consequently, the proﬁt of the manufacturer is given by Π M ≡ (w − c)(Qi + Qj ) − kθ2 .
There is a unit mass of consumers, each with a symmetric utility function: u(qi , qj , m) ≡a(qi + qj ) − b(qi2 + 2γqi qj + qj2 )/2 + m, where qi and qj are the consumption levels for products
i and j, respectively; m is the quantity of a numeraire good; γ ∈ (0, 1) is the measure of
product substitutability; and a and b are positive parameters. This utility function yields
the following demand function: qi = [a(1 − γ) − pi + γpj ]/[b(1 − γ 2 )]. As we assume that
consumers must receive the advertisement to buy the products, the demand faced by theretailers depends on θ (indicating the fraction of consumers viewing the advertisement)andequals Qi (pi , pj ) ≡ θqi and Qj (pi , pj ) ≡ θqj .3 To guarantee an interior solution, we assumek > (a − c)2 /[4b(2 + γ − γ 2 )]. This assumption is obtained from the condition that theequilibrium level of informative advertising is less than 1. Additionally, this assumption is a
suﬃcient condition for concavity of the manufacturer’s proﬁt.
We deﬁne consumer and total surpluses as follows: CS ≡ θ[a(qi + qj ) − b(qi2 + 2γqi qj +qj2 )/2 − pi qi − pj qj ], T S ≡ CS + ΠM + πi + πj .
The timing of this game is as follows: In the ﬁrst (pre-play) stage, the retailers canchoose the type of market contract (quantity or price). In the second stage, the manufacturer
chooses the level of informative advertising θ. In the third stage, the manufacturer decides
2

We assume a quadratic advertising cost. Although some previous studies have employed linear costs
and concave beneﬁts from advertising (Nakata, 2011; Shy, 1995), quadratic advertising cost is a standard
assumption in the literature (e.g., Simbanegavi, 2009; Soberman, 2004; Zhang et al., 2012; Zhang et al.,2020; Hu and Mizuno, 2021).
3
By deﬁning a utility function of the representative consumer as u(Qi , Qj , m) = a(Qi + Qj ) − b(Q2i +2γQi Qj + Q2j )/(2θ) + m, we can derive these demand functions. This demand function and set-up underinformative advertising are similar to that in Zhang et al. (2020) and Hu and Mizuno (2021).

5

the wholesale price w. In the fourth stage, the retailers compete according to the market
contract chosen in the ﬁrst stage. We solve this model using backward induction.

3

Analysis

3.1

Fourth stage: retail competition

When both retailers choose the price contract in the ﬁrst stage, in the fourth stage, theychoose pi to maximize πi = (pi − w)θ[a(1 − γ) − pi + γpj ]/[b(1 − γ 2 )]. Solving the ﬁrst-orderconditions, we obtain the prices that the retailers choose, and then the quantities as follows:pB
i (w) =

θ(a − w)a(1 − γ) + w, QB,
i (w, θ) =2−γb(2 − γ)(γ + 1)

(1)

where the superscript B denotes Bertrand competition.
Next, we consider the case in which both retailers choose quantities. Solving Qi (pi , pj ) andQj (pi , pj ) for pi and pj , we obtain the inverse demand function pi (Qi , Qj ) = a−b(Qi +γQj )/θ.
Substituting the inverse demand function into the proﬁt functions of retailers and solving theﬁrst-order conditions, we obtain the quantities chosen by the retailers and then the prices asfollows:
QC
i (w, θ) =a + w(1 + γ)
(a − w)θ
, pC,
i (w) =b(2 + γ)2+γ(2)where the superscript C denotes Cournot competition.
Finally, we examine the asymmetric case where one retailer chooses a quantity and theother chooses a price. Without loss of generality, we assume that retailer i chooses quantity
Qi , and retailer j chooses price pj . Solving Qi (pi , pj ) = θ[a(1 − γ) − pi + γpj ]/[b(1 − γ 2 )] for pi
and Qj , we obtain the following demand systems in strategic variables Qi and pj : pi (Qi , pj ) =a(1 − γ) + pj γ − bQi (1 − γ 2 )/θ, and Qj (Qi , pj ) = (a − pj )θ/b − Qi γ, respectively. Using the6

above demand systems and solving the ﬁrst-order conditions, we obtain the following results:(a − w)θ(2 − γ),b(4 − 3γ 2 )a(2 − γ − 2γ 2 + γ 3 ) + w(2 + γ − γ 2 − γ 3 )pi (w) = pCB (w) =,4 − 3γ 2(a − w)θ(2 − γ − γ 2 )Qj (w, θ) = QBC (w, θ) =,b(4 − 3γ 2 )a(2 − γ − γ 2 ) + w(2 + γ − 2γ 2 )pj (w) = pBC (w) =,4 − 3γ 2
Qi (w, θ) = QCB (w, θ) =

(3)
(4)
(5)
(6)

where the superscript CB (BC) denotes that these results are obtained when one retailer
competes in quantity (price) while the rival competes in price (quantity).

3.2

Third and second stages: manufacturer decisions

In the third stage, the manufacturer chooses the wholesale price. In the Bertrand case,2
B
maximizing ΠM = (w − c)[QBi (w, θ) + Qj (w, θ)] − kθ for w yields w = (a + c)/2. Similarly,
2
Cin the Cournot case, maximizing ΠM = (w − c)[QC
i (w, θ) + Qj (w, θ)] − kθ leads to w = (a +c)/2. In the asymmetric Cournot-Bertrand case, on maximizing Π M = (w − c)[QCB (w, θ) +QBC (w, θ)] − kθ2 , we obtain w = (a + c)/2. Therefore, we obtain the following lemma.
Lemma 1 The manufacturer chooses

w=

a+c
,
2

and the equilibrium wholesale price is independent of the type of market contract and levelof informative advertising.
Lemma 1 has a simple intuition. The manufacturer faces a linear demand, and thetype of market contract and level of informative advertising aﬀect the size of demand forthe manufacturer, but do not aﬀect the choke price. Therefore, the wholesale price thatmaximizes the manufacturer’s proﬁt takes the same value for all types of market contracts
and levels of advertising.
7

In the second stage, the manufacturer decides the level of informative advertising. In the
Bertrand case, we substitute (1) and Lemma 1 into ΠM and solve the ﬁrst-order conditionfor θ to obtain the following:θB =(a − c)2.
4kb(2 − γ)(1 + γ)

(7)

Next, in the Cournot case, we substitute (2) and Lemma 1 into Π M and solve the ﬁrst-ordercondition for θ, to obtain the following level of advertising:θC =(a − c)2.
4kb(2 + γ)

(8)

Similarly, in the asymmetric Cournot-Bertrand case, we substitute (3), (5), and Lemma 1
into ΠM , and solve the ﬁrst-order condition for θ to obtain the following:θBC = θCB =(a − c)2 (4 − 2γ − γ 2 ).
8kb(4 − 3γ 2 )(9)Comparing the advertising levels in the three cases above, we observe that θB − θBC =(a−c)2 γ 2 (2−γ −γ 2 )/[8kb(2+γ −γ 2 )(4−3γ 2 )] > 0 and θBC −θC = (a−c)2 γ 2 (2−γ)/[8kb(2+γ)(4 − 3γ 2 )] > 0, which leads to the following result.4
Lemma 2 The manufacturer chooses the level of informative advertising as θB = (a −c)2 /[4kb(2 − γ)(1 + γ)] in the Bertrand case; θC = (a − c)2 /[4kb(2 + γ)] in the Cournot case;and θBC = (a − c)2 (4 − 2γ − γ 2 )/[8kb(4 − 3γ 2 )] in the asymmetric Cournot-Bertrand case,with θB > θBC > θC .
The above results can be interpreted as follows: Following previous research (Singh andVives, 1984), we know that the market is the most competitive under price competition,
relatively less competitive under asymmetric competition, and the least competitive underquantity competition. Additionally, advertising expenditure increases with the intensity ofcompetition because the manufacturer faces greater demand in a more competitive market.
Therefore, we can infer that θB > θBC (≡ θCB ) > θC , suggesting that the manufacturer
4

Owing to 0 ≤ γ < 1, the inequality is satisﬁed.

8

advertises the most in the symmetric Bertrand competition market, relatively less in theasymmetric Cournot-Bertrand market, and the least in the symmetric Cournot competition
market.
Here, we present the proﬁts of the retailers and manufacturer as well as the consumer
surplus and total surplus in each case. Using (1)-(9) and Lemma 1, we obtain the following
outcomes. First, in the Bertrand case, we obtain the following results:(a − c)4(a − c)4 (1 − γ)
B
=
,
Π
,
M
16b2 k(1 + γ)2 (2 − γ)3
16b2 k(2 − γ)2 (1 + γ)2(a − c)4(a − c)4 (5 − 3γ)
B
CS B =,
TS=.
16b2 k(2 − γ)3 (1 + γ)2
16b2 k(2 − γ)3 (1 + γ)2πB =Next, in the Cournot case, we have the following outcomes:(a − c)4(a − c)4
C
,
Π
=
,
M
16b2 k(2 + γ)3
16b2 k(2 + γ)2(a − c)4 (1 + γ)
(a − c)4 (5 + 2γ)
CCCS =, TS =.
16b2 k(2 + γ)3
16b2 k(2 + γ)3πC =Finally, in the asymmetric case, we obtain the following outcomes:(a − c)4 (2 − γ)2 (1 − γ 2 )[4 − γ(2 + γ)](a − c)4 (2 − γ − γ 2 )2 [4 − γ(2 + γ)]
BC
,
π
=
,
32b2 k(4 − 3γ 2 )3
32b2 k(4 − 3γ 2 )3(a − c)4 [4 − γ(2 + γ)]2(a − c)4 (2 − γ 2 )[4 − γ(2 + γ)]
BC
BC
ΠM =
, CS =,
64b2 k(4 − 3γ 2 )2
64b2 k(4 − 3γ 2 )2(a − c)4 [4 − γ(2 + γ)][5 − γ(3 + γ)]
T S BC =.
32b2 k(4 − 3γ 2 )2
π CB =

3.3

First stage: endogenous market contracts

Now, we can analyze the choice of quantity versus price in the ﬁrst stage. Per the outcomesin each type of market contract, the payoﬀ matrix in the ﬁrst stage is presented in Table 1.

9Table 1: Payoﬀ matrix in the endogenous competition choice

Retailer i

Quantity
Price

Retailer j
Quantity
Price
C
C
CB
π , π
π , π BCπ BC , π CB
πB , πB

The retailers simultaneously and independently choose the type of market contract. To
identify the equilibrium contract type, we compare the retail proﬁts between π B , π C , π CB ,and π BC , and obtain the following inequalities.

π B ≤ π CB , π BC < π C if 0.392 ≤ γ < 1,π B < π CB , π C ≤ π BC if 0.378 ≤ γ < 0.392,





 π CB < π B , π C < π BC if 0 < γ < 0.378.

(10)

From the above result, we have the following proposition.
Proposition 1 In the case of informative advertising by the manufacturer, the following canbe observed: (i) When the degree of product substitutability is relatively high, i.e., 0.392 ≤γ < 1, SPNE will result in both retailers choosing to oﬀer quantity contracts (CC). (ii) Whenthe degree of product substitutability is not too high, i.e., 0.378 ≤ γ < 0.392, there are two
asymmetric SPNE at the ﬁrst stage, with one retailer oﬀering a price contract and the otheroﬀering a quantity contract (BC and CB). (iii) When the degree of product substitutability isrelatively low, i.e., 0 < γ < 0.378, SPNE will result in both retailers choosing to oﬀer pricecontracts in the ﬁrst stage (BB).
The intuition behind this proposition is as follows: There are two opposing eﬀects on theretailers’ decisions. The ﬁrst is competition eﬀect, which suggests that the adverse impact onretailers’ proﬁts intensiﬁes as the market becomes increasingly competitive. It is understoodthat the market is the most competitive under price competition, relatively less competitive
under asymmetric market conditions, and the least competitive under quantity competition.
Therefore, in the absence of informative advertising by the manufacturer, retailers are likely
10to choose quantity competition, which is conventional wisdom (Singh and Vives, 1984).
However, in our model, there is also a second eﬀect—the advertising eﬀect. From Lemma
2, the manufacturer advertises most heavily in Bertrand competition market, relatively lessin Cournot-Bertrand market, and the least in Cournot competition market. Therefore, there
will be more latent consumers who are converted to actual buyers in a Bertrand competition
market than in an asymmetric market or a Cournot competition market. More advertisingleads to greater demand for the retailers, which beneﬁts them. As the degree of substitutability falls, competition eﬀect becomes relatively unimportant and is dominated by theadvertising eﬀect. Consequently, when the degree of product substitutability is suﬃciently
low, both retailers choose prices. On the contrary, when the degree of product substitutability
is suﬃciently high, competition eﬀect becomes important and dominates advertising eﬀect,
incentivizing the two retailers to choose quantities. Finally, when the degree of productsubstitutability is not too high, as in the asymmetric Cournot-Bertrand competitive market,
which is less competitive than a Bertrand market but more competitive than a Cournot
competition market, the strategic substitutes outcomes (BC and CB) occur.
Next, we compare the manufacturer’s proﬁts, consumer surpluses, and total surpluses
under diﬀerent retailers’ market contract regimes. Owing to 1 > γ > 0, we can easily derive
BC
C
B
that ΠB> CS BC > CS C , and T S B > T S BC > T S C . Therefore, we
M > ΠM > ΠM , CS

obtain the following proposition.
Proposition 2 The proﬁt of the manufacturer, consumer surplus, and total surplus are thehighest under the Bertrand case, second highest under the Cournot-Bertrand case, and thelowest under the Cournot case.
This result is highly intuitive. Bertrand competition intensiﬁes competition and increases
informative advertising. Therefore, when both retailers choose prices, the problem of double
marginalization becomes less important, and many consumers can buy the products. Consequently, the manufacturer’s proﬁt, consumer surplus, and total surplus are the highest under
Bertrand competition. By contrast, Cournot competition is the least competitive and oﬀers
the least informative advertising. The asymmetric case lies between Bertrand competition
11and Cournot competition in terms of the intensity of competition, which leads to the above
result.
It is a well-known result that while Cournot competition is the optimal choice for ﬁrms,
consumer and total surpluses are invariably the largest under Bertrand competition. However, in our model, Bertrand competition could occur in equilibrium and simultaneously
beneﬁt both consumer and total surpluses.

4

Discussion

4.1

Timing of advertising

In the previous section, we assumed that retailers choose the market contract before informative advertising. In this subsection, we examine whether the main result of Proposition
1 still holds if the timing of choosing the market contract is changed.
Hence, we consider the following two types of timings: (i) In the ﬁrst stage, the manufacturer chooses the level of advertising; in the second stage, the manufacturer sets its wholesale
price; in the third stage, the retailers choose the market contract (quantity or price); and in
the fourth stage, following the market contracts determined in the third stage, the retailers
compete with each other. (ii) In the ﬁrst stage, the manufacturer chooses the level of advertising; in the second stage, the retailers choose the market contract; in the third stage, themanufacturer sets the wholesale price; and in the fourth stage, the retailers compete on thevariable chosen in the second stage.
First, we consider case (i). Outcomes in the fourth stage are the same as those obtainedin the previous section, (1)–(6). Then, the proﬁts of retailers under each case are as follows:(a − w)2 (1 − γ)θ
(a − w)2 θ
CT
,
π
=
,
b(2 − γ)2 (1 + γ)b(2 + γ)2(a − w)2 (2 − γ − γ 2 )2 θ
(a − w)2 (2 − γ)2 (1 − γ 2 )θ
BCT
,
π
=
.
π CBT =b(4 − 3γ 2 )2b(4 − 3γ 2 )2π BT =

where the superscript BT , CT , and CBT (BCT ) denote—all with reversed timing of the12

game—Bertrand competition, Cournot competition, and the asymmetric case obtained when
the retailer competes in quantity (price) while the rival competes in price (quantity), respectively. Comparing the above proﬁts, we obtain π CT > π CBT > π BT > π BCT . Hence, in any
subgame, the retailers prefer choosing a quantity contract, which results in a Cournot game
(CC). This result is the same as the well-known result of Singh and Vives (1984). Our main
result is that the case wherein both retailers choose price (BB), and the asymmetric casewherein one retailer chooses price and the other chooses quantity (BC and CB) never occur
in equilibrium.
Next, we consider case (ii). In the third stage, the manufacturer chooses a wholesale
price. From Lemma 1, the manufacturer chooses w = (a + c)/2 in any market contract.
Additionally, with case (i), we already revealed that for any wholesale price w, the retailers
prefer choosing a quantity contract. Consequently, both retailers choose quantities (CC).
Importantly, this result proves that the result of Singh and Vives (1984) still holds in a
vertical structure with an upstream manufacturer setting the wholesale price, but it does
not hold when informative advertising by the manufacturer is introduced.
Summarizing the results in the two cases, we ﬁnd that the assumption that the retailers
choose the market contract before informative advertising is crucial for our main results.

4.2

Persuasive advertising

While in the previous section we considered informative advertising, in this subsection, we
discuss persuasive advertising, which increases the willingness to buy or marginal utility for
consumers.
Model We assume that the utility function of the representative consumer is u = (a +θ)(Qi +Qj )−b(Q2i +2γQi Qj +Q2j )/2+m, where θ denotes the level of persuasive advertising.5
Then, the demand for each product is Qi = [(a + θ)(1 − γ) − pi + pj γ]/[b(1 − γ 2 )]. The proﬁts
of the manufacturer and retailer i are ΠM = (w − c)(Qi + Qj ) − kθ2 and πi = (pi − w)Qi ,5

The utility function under persuasive advertising is similar to Zhang et al. (2020), and Hu and Mizuno(2021).

13

respectively. The other settings are the same as those in the previous section. For concavityof the manufacturer’s proﬁt, we assume bk ≡ z > (1 + a − c)/[2(2 + γ − γ 2 )] ≡ z SOC .
Calculating equilibrium In the fourth stage, when both ﬁrms choose prices, on substituting the demand function into the retailers’ proﬁt function and maximizing it with respect
to the price, we obtain the following price and quantity:
pBpa(w, θ) =ia−w+θ
(1 − γ)(a + θ) + w, QBpa(w, θ) =,i
2−γb(2 + γ − γ 2 )where the superscript Bpa denotes Bertrand competition with persuasive advertising.
Next, we consider the case when both retailers choose quantities. Solving Qi = [(a +θ)(1 − γ) − pi + pj γ]/[b(1 − γ 2 )] for pi and pj , we can obtain the inverse demand functionpi (Qi , Qj ) = a+θ −b(Qi +γQj ). Using this inverse demand function, we obtain the following
quantities and prices in this subgame.

QCpa(w, θ) =ia + θ + w(1 + γ)
a−w+θ
, pCpa(w, θ) =,ib(2 + γ)2+γwhere the superscript Cpa denotes Cournot competition with persuasive advertising.
Finally, we analyze the asymmetric case wherein one retailer chooses quantity and theother chooses price. Without loss of generality, we assume that retailer i chooses quantity
Qi , and retailer j chooses price pj . Solving the demand function Qi = [(a + θ)(1 − γ) −pi + pj γ]/[b(1 − γ 2 )] for pi and Qj , we obtain the demand systems in strategic variables
Qi and pj : pi (Qi , pj ) = (a + θ)(1 − γ) + pj γ − bQi (1 − γ 2 ), and Qj (Qi , pj ) = (a + θ − pj −bQi γ)/b, respectively. Using the above demand systems and solving the ﬁrst-order conditions14∂πi (Qi , pj )/∂Qi = 0 and ∂πj (Qi , pj )/∂pj = 0, we obtain the following quantities and prices:(a − w + θ)(2 − γ),b(4 − 3γ 2 )(a + θ)(2 − γ − 2γ 2 + γ 3 ) + w(2 + γ − γ 2 − γ 3 )pi (w, θ) = pCBpa (w, θ) =,4 − 3γ 2(a − w + θ)(2 − γ − γ 2 )Qj (w, θ) = QBCpa (w, θ) =,b(4 − 3γ 2 )(a + θ)(2 − γ − γ 2 ) + w(2 + γ − 2γ 2 )pj (w, θ) = pBCpa (w, θ) =4 − 3γ 2
Qi (w, θ) = QCBpa (w, θ) =where the superscript CBpa (BCpa) indicates that this result is obtained when a retailer
competes in quantity (price) while its rival competes in price (quantity) with persuasive
advertising.
In the third stage, the manufacturer chooses its wholesale price. Regardless of whichmarket contract regime the retailers choose, the manufacturer chooses the same wholesale
price: w = (a + θ + c)/2.
In the second stage, for the case wherein both retailers choose prices, the manufacturer
chooses the following:θBpa =a−c.
2z(2 + γ − γ 2 ) − 1

Note that z ≡ bk. Then, for the case wherein both retailers choose quantities, we derive thefollowing level of advertising:θCpa =a−c.
2z(2 + γ) − 1

For the asymmetric case wherein one retailer chooses quantity and the other chooses price,
we obtain the following:θBCpa = θCBpa =(a − c)(4 − 2γ − γ 2 ).
4z(4 − 3γ 2 ) + 2γ + γ 2 − 415Comparing the advertising levels in the three cases above, we observe the following:
2z(a − c)γ 2 (2 − γ − γ 2 )> 0,[1 − 2z(2 − γ)(1 + γ)][4 − γ(2 + γ) − 4z(4 − 3γ 2 )]2z(a − c)γ 2 (2 − γ)θBCpa − θCpa => 0.
[1 − 2z(2 + γ)][4 − γ(2 + γ) − 4z(4 − 3γ 2 )]

θBpa − θBCpa =Notably, the assumption z > z SOC ensures that the denominators are positive. Hence, we
obtain the following lemma with an intuition similar to Lemma 2.
Lemma 3 The level of persuasive adverting is ranked as follows: θBpa > θBCpa = θCBpa >θCpa .
Here, substituting the subgame outcomes into the proﬁts of retailers, we obtain theretailer’s proﬁts as follows:ξ(1 − γ 2 )ξ, π Cpa =,22[1 − 2z(2 + γ − γ )]1 − 2z(2 + γ)]2
4ξ(2 − γ − γ 2 )2
4ξ(2 − γ)2 (1 − γ 2 )
BCpa
,
π
=
,
π CBpa =[4 − 2γ − γ 2 + 4z(4 − 3γ 2 )]2[4 − 2γ − γ 2 + 4z(4 − 3γ 2 )]2
π Bpa =

where ξ ≡ (a − c)2 bk 2 > 0.
Now, we consider the ﬁrst stage. Comparing the retailers’ proﬁts in the above cases, we
present the ranking for the same.
 π CBpa < π Bpa , π Cpa < π BCpa if z SOC < z < 1/(4γ),
 π Bpa < π CBpa , π BCpa ≤ π Cpa if 1/(4γ) ≤ z.
This result directly leads to the following proposition:
Proposition 3 If a manufacturer engages in persuasive advertising, when the advertising
investment is suﬃciently eﬃcient (small k) or the degree of product substitutability is sufﬁciently low (small γ)—that is, z ≡ bk < 1/(4γ), the retailers oﬀer a price contract (BB);otherwise, they oﬀer a quantity contract (CC). The asymmetric regime wherein one retailer
chooses price and the other chooses quantity never occurs.
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The intuition is as follows: When k is small, the advertising investment is eﬃcient, whichstrengthens the advertising eﬀect. When this advertising eﬀect is suﬃciently large becauseof eﬃcient advertising, it may dominate the competition eﬀect, which inﬂuences retailers
to choose a more competitive market contract (BB). Meanwhile, if the degree of productsubstitutability is suﬃciently low, similar to Proposition 1, competition eﬀect becomes lessimportant and may be dominated by advertisement eﬀect; this also leads both ﬁrms to oﬀer
a price contract (BB). Similarly, when investment is ineﬃcient with large k or the degreeof product substitutability is suﬃciently large, retailers oﬀer a quantity contract (CC). The
incentives to oﬀer a price contract (or quantity contract) diﬀer depending on the type ofcontract chosen by the rival retailer. Hence, there are two thresholds for oﬀering a pricecontract, and in this study, those thresholds are equal. Consequently, each ﬁrm chooses acontract type based on its dominant strategy.

5

Conclusions

We consider a supply chain with a manufacturer engaged in advertising and two retailers producing diﬀerentiated products. We analyze the endogenous choice between quantity or pricecompetition for retailers. We ﬁnd that depending on the degree of product substitutability,
the set of possible outcomes is rich and includes Cournot, Bertrand, and Cournot-Bertrand
under informative advertising, which contrasts with the well-known result that the dominant
strategy is for ﬁrms to compete à la Cournot. Moreover, we demonstrate that under bothinformative and persuasive advertising, the manufacturer, consumer, and total surpluses are
the largest when retailers oﬀer a price contract. Additionally, under persuasive advertising,either Cournot or Bertrand outcomes may occur in equilibrium, but Cournot-Bertrand neveroccurs.
It is worthwhile to consider wholesale price discrimination and also advertising diﬀerentiation, wherein the manufacturer oﬀers diﬀerent levels of advertising to the two retailers.
We leave this investigation for future research.

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