D B S －１８－０１

A Differentiated Oligopoly in Which Every Firm
Welcomes Tougher Competition
Tatsuhiko Nariu

Graduate School of Business, Doshisha University, JapanDavid Flath

Faculty of Economics Ritsumeikan University, Japan

Makoto Okamura Economics Department, Gakusyuin University, Japan2018 年 9 月

We offer an example in which increased competition (softening of productdifferentiation) increases the profit of all incumbent firms.
Key Words: manufacturer-suppliercomponents,
vertical strategic effectrelation,

productdifferentiation,

JEL Classification: D43, L13

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A Differentiated Oligopoly in Which Every Firm Welcomes
Tougher Competition
1. Introduction
Increased market competition—defined as any exogenous factor that lowers
equilibrium prices without shifting the scale of demand—is generally presumed todecrease profit. Here, “increased competition” could mean entry of new firms or softeningof product differentiation. Some recent papers have qualified this presumption thatincreased competition harms profit, by offering examples in which increased market
competition actually increases the profit of at least one incumbent firm …but not allincumbent firms. We offer an example in which increased competition (softening ofproduct differentiation) does increase the profit of all incumbent firms.
Ishida et. al. (2011) present an asymmetric Cournot oligopoly model in which
each firm has a constant marginal cost that it can lower by investing in R&D. When afirm with high marginal cost enters the market it can stimulate R&D investment by a lowcost firm by enough to increase the profit of that low-cost firm compared to the pre-entry
equilibrium. However, incumbent firms with high marginal cost suffer losses. Mukherjeeand Zhao (2009) show that entry by a Stackelberg follower with high marginal cost canincrease the profit of the more efficient Stackelberg leader. Matsushima and Mizuno(2010) develop an example in which stronger upstream competition (introducing a rival
into the upstream) actually increases the profit of an incumbent upstream firm. CorreaLópez (2007) shows that when downstream firms change from quantity competition toprice competition, their prices fall but their profits can increase because the wage set bynegotiation between the firm and its labor union becomes low. The upstream suppliers(the labor unions) are worse off in the end. Finally, Fanti (2013) analyses a duopoly modelwith partial cross ownership and wage negotiation between each firm and its labor union,and shows that profits of both firms can increase as their products become morehomogenous, but again the upstream labor union is worse off. In all of these examples,increased competition benefits some firms. We offer an example in which it benefits allfirms.
In our example, the profit of every firm in a vertical industry increases as theproducts become more homogenous. In the example, each of multiple manufacturers
forms a differentiated final product by assembling n components that are provided from1its own n independent suppliers. Each supplier produces a unique component and sells to
only one of the manufacturers. The multiple manufacturers attain a product-differentiated
Cournot oligopoly equilibrium. We will show that if the products are sufficiently
homogeneous to begin with and the number of suppliers is large enough, then every
firm—including manufacturers and suppliers—earns more profit as the final goods
become more homogenous.
2. The Model
Suppose that 𝑚(𝑚 ≧ 2) competing manufacturers i (=1,..,m) with the same
Leontief technology, each obtains from its own suppliers the n components (inputs)
needed to assemble a final product. Each supplier ij (j=1,...,n) provides a different
component to manufacturer i. Each component is produced at zero marginal cost. No
additional cost is required for production of the final goods.
The inverse demand function facing each manufacturer i is given by𝑝𝑖 = 𝑎 − 𝑞𝑖 − 𝑏  q h , 𝑖 = 1, … , 𝑚; ℎ = 1, … , 𝑛, and 𝑖 ≠ ℎ,(1)h iwhere 𝑝𝑖 stands for the price of good i , 𝑞𝑖 is the supply of good i , the parameter 𝑎 >0 shows the market size, and 𝑏 ∈ (0,1) shows the degree of product differentiation.
We will consider the following game:Stage 1: Each supplier ij sets its component price 𝑤𝑖𝑗
Stage 2: Each manufacturer i, after observing the component prices of all suppliers—itsown suppliers and those of its rivals—decides its output quantity 𝑞𝑖 and purchases fromits suppliers the components needed to produce such quantity. The outputs of the 𝑚manufacturers are sold at the market-clearing prices.
3. The Sub-game Perfect Equilibrium.
We derive the sub-game perfect equilibrium of the game by backward induction.
In the second stage, given the component prices 𝑤𝑖𝑗 , each manufacturer i determines its
supply 𝑞𝑖 to maximize its profit
yi   a  qi  b qh   wij qihij(2)where  wij  ci is the marginal cost of manufacturer i. By the profit-maximizingj2

condition, the equilibrium output of manufacturer i in the second-stage game is given by(2  b) a   wij   b wijjh i j,qi ( w ) (2  b)(2  bm  1)(3)

where 𝒘 is the vector whose elements 𝑤𝑖𝑗 are the shipping prices of the nm componentsuppliers. One may interpret Eq. (3) as the derived demand facing the n suppliers ofcomponents to manufacturer i.
From (3), the derived demand function facing the ij component supplier,qi wi j : wij

 where 𝒘−𝑖𝑗 denotes the component price vector excluding 𝑤𝑖𝑗 , isnegatively sloped with respect to 𝑤𝑖𝑗 . Moreover,𝜕 2 𝑞𝑖=−𝜕𝑤 𝜕𝑏𝑖𝑗𝑏(4+𝑏(𝑚−2))(𝑚−1)
𝐷0 2< 0,(4)

where 𝐷0 = 2 + 𝑏(𝑚 − 1) > 0. As the final products become more homogenous, theslope of the derived demand becomes steeper. We also have that if marginal cost, w  c , is the same for all manufacturers i (a condition which is satisfied at subiji

jgame perfect equilibria), then ∂𝑞𝑖 /𝜕𝑏 < 0. Accordingly, as the final products become
closer substitutes, the derived demand for components shifts downward and becomes
more elastic at each price.
The equilibrium final price and profit of manufacturer i are as follows:pi ( w ) (2  b)a  (2  b2 ) j w1 j  b wij
h i

4b2



 (2  b)a   wij  b whj jj
yi ( w )  
2 2(4  b )j(5-1)2(5-2)In the first stage, anticipating the second-stage equilibrium, each supplier ij sets itscomponent price 𝑤𝑖𝑗 to maximize its profit 𝑧𝑖𝑗 , given the component prices of the othersuppliers𝑧𝑖𝑗 = 𝑤𝑖𝑗 𝑞𝑖 ( w ) 𝑗 = 1, … 𝑛
By the profit-maximizing condition, we obtain the reaction function of component
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supplier ij,𝑤𝑖𝑗 (𝒘−𝒊𝑗 )  (2  b)a  b  whj  /( 4  2b(m  2))   wij / 2 .
h i
jj i



(6)

From (6), we find that the component price of each supplier is a strategic substitute forthe component prices of the suppliers servicing the same manufacturer, while it is astrategic complement with the component prices of the suppliers servicing the rival
manufacturers. We can infer that as the final goods becomes less differentiated (themarket competition becomes tougher), each supplier reduces its component price:𝜕𝑤𝑖𝑗 ⁄𝜕𝑏 =   (a   whj ) /( 2  b(m  2))2 <0.
hj

From the nm reaction functions similar to that of supplier 𝑖𝑗 shown by (6), we
derive the perfect-equilibrium component price which is the same for every component
supplied to every manufacturer:𝑤∗ =(2−𝑏)𝑎𝐷1,(7)where 𝐷1 = 𝑏(𝑚 − 1) + (2 − 𝑏)(𝑛 + 1) > 0. The equilibrium values of other variables
are𝑞 ∗ = (2 + 𝑏(𝑚 − 2))𝑎/(𝐷0 ∙ 𝐷1 )(8-1)

𝑝∗ = {(2 + 𝑏(𝑚 − 2)) + (2 − 𝑏)(2 + 𝑏𝑛(𝑚 − 1))}𝑎/(𝐷0 ∙ 𝐷1 )(8-2)

𝑐 ∗ = 𝑛𝑤 ∗ = 𝑛(2 − 𝑏)𝑎/𝐷1(8-3)

𝑦 ∗ = 𝑞2

(8-4)

𝑧 ∗ = (2 − 𝑏)(2 + 𝑏(𝑚 − 2))𝑎2 /(𝐷0 2 ∙ 𝐷1 2 )(8-5)

Each manufacturer’s price-cost margin, 𝜃 ∗ = 𝑝∗ − 𝑐 ∗ , is equal to its output, 𝑞 ∗ .
4. Comparative Statics and Results
In this section, based on the comparative static analysis of the perfect equilibrium,
we derive our main results.
By differentiating 𝑞 ∗ and 𝑝∗ with respect to the parameter, 𝑏, we find that as thegoods become more homogeneous, the inverse residual demand for each final good,pi  (a  b q j )  qi , shifts downward and each manufacturer decreases its supply.
j iOwing to the downward shift in demand, each component supplier decreases its price,4𝜕𝑤 ∗𝜕𝑏= −2(𝑚 − 1)⁄𝐷1 2 < 0, implying that the marginal cost of every manufacturer, 𝑐 ∗ ,also decreases. Moreover, we have𝜕 2 𝑤 ∗ 4(𝑚 − 1)(𝑚 − 𝑛 − 2)
=
⋚ 0,
𝜕𝑏 2
𝐷1 2

iff 𝑛 ⋚ 𝑚 − 2.

That is, when 𝑛 > 𝑚 − 2, as b becomes larger, the component prices decrease by everincreasing decrements, and so each manufacturer’s marginal cost decreases by everincreasing decrements. To put it another way, as b becomes larger (the products ofdiffering manufacturers become more homogeneous), the vertical strategic effectsbecome stronger. Here, “vertical strategic effect” means that as manufacturer i supplies
less output, it induces suppliers of components to its rival manufacturers to raise theirprices, increasing the marginal costs of the rivals. As b and n become large the verticalstrategic effects become great enough to outweigh the horizontal strategic effects that arealso present, and the price-cost margin 𝜃 ∗ increases. The horizontal strategic effect isthat as a Cournot manufacturer supplies more output, its rivals react by suppling lessoutput.
Differentiating the manufacturer price-cost margin 𝜃 ∗ with respect to b, we have𝜕𝜃 ∗ (𝑚 − 1)𝑏 2 (𝑚 − 2)(𝑛 − 𝑚 + 2) − 4(1 − 𝑏)=.
𝜕𝑏𝐷1 2𝜕𝜃∗

While the sign of 𝜕𝑏 is negative if n<m-2, in the case where n>m-2
𝜕𝜃 ∗⋛0𝜕𝑏iff𝑏⋛2= 𝑏0 .
𝑛 − 𝑚 + 2 + 𝑛(𝑛 − 𝑚 + 2)1⁄2

That is, 𝜃 ∗ (𝑏) is a decreasing (increasing) function in the interval 𝑏 < 𝑏0 (𝑏 > 𝑏0), andattains its minimum value at 𝑏 = 𝑏0.

Noting that 𝑛 >𝑚2𝑚+2≡ 𝑛0 is needed for 𝑏 < 1,we have
Lemma 1: If 𝑛 > 𝑛0 , the equilibrium manufacturer price-cost margin function 𝜃 ∗ (𝑏)is single-caved in the interval 𝑏 ∈ (0,1) and attains its minimum value at 𝑏𝑜 ∈ (0,1).
Noting that 𝑞 ∗ = 𝜃 ∗ and 𝑦 ∗ = 𝜃 ∗ 𝑞 ∗ , from Lemma 1, the equilibrium profit function 𝑦 ∗is also single-caved in the interval 𝑏 ∈ (0,1) . Hence, the manufacturer’s profit attains a5

maximum either at 𝑏 = 0 or 𝑏 = 1. Denoting the equilibrium profit with given 𝑏 and𝑛 by 𝑦 ∗ (𝑏, 𝑛), we have
𝑚𝑎𝑚𝑎y ∗ (1, 𝑛) = {(𝑚+1)(𝑚+𝑛)}2 > {(𝑚+1)(𝑚+𝑛)}2 = 𝑦 ∗ (0, 𝑛) if 𝑛 > 𝑚.
Therefore, the following proposition is established.
Proposition 1: When 𝑏 ≥ 𝑏0 , each manufacturer’s profit increases as the goods becomeless differentiated. Moreover, when 𝑏 ≥ 𝑏0 and 𝑛 > 𝑚, the profit attains a maximum
when the goods are completely homogeneous, 𝑏 = 1.
Next, we examine the profits of the suppliers. Differentiating each supplier’s profit
𝑧 with respect to 𝑏, we have𝜕𝑧 ∗ (𝑚 − 1)𝑔(𝑏, 𝑛)=,𝜕𝑏
𝐷0 2 𝐷1 3
where 𝑔(𝑏, 𝑛) =(2 − 𝑏)𝑏(4 + 𝑏(𝑚 − 2))𝑛 + 𝑏(𝑚 − 2)(2((5 − 𝑏)𝑏 − 8) − (4 − 𝑏)𝑚) − 16.
Notice that 𝑔(𝑏, 𝑛) is a cubic polynomial in b, its third-degree coefficient isnegative, and 𝑔(0, 𝑛) = −16 < 0.

Hence, if 𝑔(1, 𝑛) > 0, there is only one root, 𝑏1 ,in the interval 𝑏 ∈ (0,1) . A sufficient condition for 𝑔(1, 𝑛) > 0 is 𝑛 > 𝑛1 ≡𝑚(3𝑚 + 2)⁄(𝑚 + 2).
Lemma 2: If 𝑛 > 𝑛1 , the equilibrium profit function 𝑧 ∗ is single-caved in the interval𝑏 ∈ (0,1) and attains its minimum at 𝑏1 ∈ (0,1).
Consider the case in which the goods are sufficiently homogeneous. Each supplierreduces its component price a bit as 𝑏 increases. When in this case there are many
suppliers, each manufacturer’s marginal cost, 𝑐 ∗ = 𝑛𝑤 ∗ , decreases with 𝑏. This induceseach manufacturer to increase its output (increase the derived demand facing eachcomponent supplier), implying that each component supplier’s profit, 𝑧 ∗ = 𝑤 ∗ 𝑞 ∗ , alsoincreases. Denoting the equilibrium component supplier’s profit with given 𝑏 and 𝑛 by𝑧 ∗ (𝑏, 𝑛), we have𝑚𝑧 ∗ (1, 𝑛) = (𝑚+1)(𝑚+𝑛)2 >(𝑛+1)22= 𝑧(0, 𝑛), if 𝑛 > 𝑛2 ≡ 𝑚 + (2𝑚2 + 2𝑚)1/2
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Then we obtain
Proposition 2: When 𝑛 > 𝑛1 and𝑏 ≥ 𝑏1 ∈ (0,1), each component supplier’s profit

increases as the goods become less differentiated. If in this situation, 𝑛 > 𝑛2 , and thecomponent supplier profit attains its maximum when the goods are completely
homogeneous, 𝑏 = 1.
Here, we will show that 𝑏1 > 𝑏0 holds. Suppose not, i.e. suppose 𝑏0 ≧ 𝑏1 . Then,𝜕𝑦(𝑏1 )⁄𝜕𝑏 ≦ 𝜕𝑦(𝑏0 )⁄𝜕𝑏 = 0 holds. Because 𝜕𝑧⁄𝜕𝑏 = (𝜕𝑤⁄𝜕𝑏) ∙ 𝑞 + (𝜕𝑧⁄𝜕𝑏) ∙ w,
we have
0 = 𝜕𝑧(𝑏1 )⁄𝜕𝑏 = (𝜕𝑤(𝑏1 )⁄𝜕𝑏) ∙ 𝑞 + (𝜕𝑞(𝑏1 )⁄𝜕𝑏) ∙ 𝑤 < 0, (∵ 𝜕𝑤(𝑏1 )⁄𝜕𝑏 < 0)which results in contradiction.
Finally, we establish our main result.
Proposition 3: Suppose that the number of component suppliers is larger thanmaxn1, n2 . When 𝑏 ≥ 𝑏1, as the goods becomes less differentiated, 𝑏 → 1, the profit ofevery manufacturer and of every supplier increases. Moreover, the profit of each agent ismaximal when the goods are completely homogeneous, 𝑏 = 1.
Note that 𝑛 > 𝑛1 is a sufficient condition but not a necessary condition for the existenceof an interval for which z * b  0 . When 𝑔(1, 𝑛) < 0, there is no root in the interval𝑏 ∈ (0,1), or there are two roots, 𝑏1 and 𝑏2 . In the former case z * b  0 in theinterval 𝑏 ∈ (0,1), and in the latter case z * b  0 in the interval 𝑏 ∈ (𝑏1 , 𝑏2 ).
4. Concluding Remarks
Increased competition means anything that lowers price while holding the scale ofdemand unchanged. A widely held presumption is that increased competition lowers theprofits of at least some incumbent firms. We have constructed an example in which, on
the contrary, increased competition—meaning reduced product differentiation—
increases the profit of every incumbent upstream and downstream firm in a symmetric
Cournot oligopoly supplied by independent producers of components. This seems to be
the first example in the literature in which all incumbent firms welcome greatercompetition.
7References
Correa-López, M. (2007) “Price and Quantity Competition in a Differentiated Duopolywith Upstream Suppliers,” Journal of Economics and Management Strategy, Vol.16,
No.2, pp.469-505.
Fanti, L. (2013) “Cross-ownership and Unions in a Cournot Duopoly: WhenProfits Reduce with Horizontal Product Differentiation,” Japan and theWorld Economy, 27, pp. 34-40.
Ishida, J., T. Matsumura, and N. Matsushima (2011) “Market Competition, R&D andFirm Profit in Asymmetric Oligopoly,” Journal of Industrial Economics, Vol.59, No.3,
pp.484-505.
Matsushima, N. and Mizuno, T. (2012) “Profit-enhancing Competitive Pressure in
Vertically Related Industries,” Journal of Japanese and International Economies,
Vol.26, No.1, pp.142-152.
Mukherjee, A., and Zhao, L (2009), “Profit Raising Entry,” Journal of IndustrialEconomics, Vol.57, No.4, pp.870-870.

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