A note on monopolistic competition with OEMToshihiro Atsumi
Faculty of Economics, Meiji Gakuin UniversityDiscussion Paper No.22-01
2023 年 2 月

A note on monopolistic competition with OEMAbstract
This paper studies a kind of original equipment manufacturing (OEM) defining it as a mode ofbusiness in which new firms can enter by asking an incumbent firm to supply an OEM productwhich will be sold to the consumers under its brand name. It is found that the existence of OEM
can be explained as an equilibrium outcome, in a form of multiple equilibria. There are no welfareeffects in comparison to the standard monopolistically competitive equilibrium. The production
structure, however, becomes lumpy under the OEM equilibria: there will be larger producers acting
as manufacturers of their own brands as well as OEM suppliers. Firm heterogeneity arises not fromex-ante productivity difference but from OEM.

1. Introduction
Original equipment manufacturing (OEM) is a popular business practice. I define OEM as a modeof business in which new firms can enter by asking an incumbent firm to supply an OEM productwhich will be sold to the consumers under its brand name. Although there are other types of OEM,I focus on this type of OEM.
The idea of this note is to consider OEM as an alternative form of firm entry in the framework ofmonopolistic competition in general equilibrium. I study whether OEM can be explained as anequilibrium outcome.
It is found that the existence of OEM can be explained as an equilibrium outcome, in a form ofmultiple equilibria. There are no welfare effects in comparison to the standard monopolisticallycompetitive equilibrium, as long as the varieties including OEM varieties are valued equally byconsumers. The production structure, however, becomes lumpy under the OEM equilibria: there
will be larger producers acting as manufacturers of their own brands as well as OEM suppliers.
Therefore, firm heterogeneity arises not from ex-ante productivity difference but from OEM.
In the next section I review the standard general equilibrium results of monopolistic competition.
In section 3, I derive conditions under which a potential entrant uses OEM to set up its own brand.
Section 4 studies equilibrium with OEM, followed by concluding comments.

2. Standard monopolistically competitive equilibrium
Assumptions
Consumer preference is given by𝑛1

𝑈 = [∫0 𝑚(𝑖)𝜌 𝑑𝑖 ]𝜌 (0 < 𝜌 < 1),(1)which is a constant elasticity of substitution (CES) function, and 𝑚(𝑖) is the consumption of eachvariety 𝑖, and 𝑛 is the total mass of variety. The elasticity of substitution between any variety is1⁄(1 − 𝜌) ≡ 𝜎 (𝜎 > 1).
On the production side, a firm producing a particular variety requires a fixed number (𝐹) of workers
and 𝑐 units of workers per unit output. The firm thus face s increasing returns to scale. Its total costfor producing a given amount, 𝑞 is then𝐶(𝑞) = 𝐹𝑤 + 𝑐𝑤𝑞,(2)
where 𝑤 is the wage of the workers. It is assumed that this industry is monopolistically competitive.
Factor endowment, that is, total workers, is 𝐿, and the workers are also consumers.
Consumer behavior
For a given income 𝑌, and a given price 𝑝(𝑖) for each variety, the consumers problem is tomaximize her utility, subject to the budget constraint𝑛(3)∫0 𝑝(𝑖)𝑚(𝑖)𝑑𝑖 = 𝑌.
Then consumers should choose 𝑚(𝑖) to minimize the cost of consuming 𝑈 . This implies
minimizing expenditure1𝑛∫ 𝑝(𝑖)𝑚(𝑖)𝑑𝑖0

subject to1𝑛[∫0 𝑚(𝑖)𝜌 𝑑𝑖 ]𝜌 = 𝑈.
(4)
The first-order condition of the problem is that the marginal rate of substitution between any twovarieties 𝑖 and 𝑗 is equal to their price ratio:𝑚(𝑖)𝜌−1𝑚(𝑗)𝜌−1=𝑝(𝑖)

.

(5)

𝑝(𝑗)

Substituting (5) into (4) produces the following expression, which is the compensated demand
function for variety 𝑗:
1𝑚(𝑗) =𝑝(𝑗)𝜌−11𝜌𝜌𝑛𝜌−1[∫0 𝑝(𝑖)
𝑑𝑖]

𝑈.

(6)

The minimum expenditure to consume 𝑈 is, therefore,
𝜌

𝜌−1

𝜌𝑛𝑛
∫0 𝑝(𝑗)𝑚(𝑗)𝑑𝑗 = [∫0 𝑝(𝑖)𝜌−1 𝑑𝑖 ]

𝑈.

(7)

The term𝑛𝜌𝜌−1𝜌

[∫ 𝑝(𝑖)𝜌−1 𝑑𝑖 ]0in (7) is called the price index, 𝐺:𝜌𝜌−1𝜌𝑛𝐺 ≡ [∫0 𝑝(𝑖)𝜌−1 𝑑𝑖]Using 𝐺, (6) can be written simply as1𝑛= [∫0 𝑝(𝑖)1−𝜎 𝑑𝑖]1−𝜎 .
𝑝(𝑗) −𝜎𝑚(𝑗) = [ ] 𝑈.
𝐺Then the demand function for variety 𝑗 is𝑚(𝑗) = 𝑝(𝑗)−𝜎 𝐺 𝜎−1 𝑌.

(8)
(9)
(10)

Using (10), price elasticity of demand[𝑑𝑚(𝑗)⁄𝑑𝑝(𝑗)]⁄[𝑚(𝑗)⁄𝑝(𝑗)]becomes𝑛𝜎 + [𝑝(𝑗)1−𝜎 (1 − 𝜎)]⁄∫0 𝑝(𝑗)1−𝜎 𝑑𝑗.
(11)
Since 𝑛 → ∞ under a continuum of varieties, the second term approaches zero. The price elasticity
of demand is therefore simply 𝜎.
Firm behavior
Because of the infinite number of potential varieties and increasing returns to scale at the firm level,
each firm becomes a sole producer of differentiated variety. The first-order condition of profit
maximization is then the equalization of marginal revenue and marginal costs. Since the demandelasticity that each firm faces is 𝜎, firms will exhibit the following mark-up pricing behavior
1𝑝 (1 − ) = 𝑐𝑤,(12)
𝜎2and the price index of goods is1

1

𝐺 = [𝑛𝑝1−𝜎 ]1−𝜎 = 𝑛1−𝜎 𝑝.

(13)Equilibrium
Equilibrium requires market clearing in the goods and factor market, and zero profit of firms as aresult of free entry. Then in equilibrium, operating profits equals fixed cost. That is
𝑝𝑞 − 𝑐𝑤𝑞 = 𝐹𝑤.
(14)
Another equilibrium condition is clearing of the labor market. Since each firm employs 𝐹 + 𝑐𝑞
workers, full employment requires𝑛(𝐹 + 𝑐𝑞) = 𝐿.
(15)
Finally, goods markets need to clear in equilibrium, that is,
𝑝−𝜎 𝐺 𝜎−1 𝑌 = 𝑞.
(16)
Substituting for 𝑝 in (14) using (12) and rearranging terms produces a constant equilibrium firmsize:
𝐹(𝜎−1)𝑞=.
(17)𝑐
Substituting (17) into (15) and rearranging terms results in a equilibrium firm mass of𝐿
𝑛= ,
(18)
𝐹𝜎
And it can be confirmed that (17) and (18) satisfy (16).

3. Entry by use of OEM
In the standard monopolistically competitive equilibrium, each firm produces its own product tobe sold under its brand name. The idea of the analysis is to allow for entry by use of OEM. A
potential entrant considers profitability of setting up a new brand using OEM. That is, the potential
entrant, instead of manufacturing a new variety on its own, asks an incumbent firm to supply
products to be sold under this potential entrant’s brand name.
Suppose the OEM product is available at a unit price of 𝑐 ∗ , by setting 𝑤 = 1. Then the potential
entrant can calculate its hypothetical profit 𝜋 ∗ = 𝑝∗ 𝑞 ∗ − (𝐹 ∗ + 𝑐 ∗ 𝑞 ∗ ), where𝜎𝑐 ∗𝑝∗ =(19)𝜎−1and𝑞 ∗ = 𝑝∗ −𝜎 𝑛−1 𝑝𝜎−1 𝑌.
(20)
Hereafter, variables given asterisks (*) are those for the OEM firms. Using (19) and (20),𝑐∗𝜎𝑐 ∗ −𝜎 𝐹𝜎𝜎 𝜎−1∗𝜋 =()𝐿)() ( )(𝜎−1 𝜎−1𝐿
𝜎−1
Therefore the potential entrant will decide to enter as an OEM firm if 𝜋 ∗ > 0, or𝑐 𝜎−1𝐹∗( ∗)> ,(21)𝑐𝐹∗
where 𝑐 is the unit price offered by an incumbent firm that manufactures on its own, or in-housemanufacturing (hereafter IHM) firm, to the OEM entrant, and 𝐹 ∗ is the fixed cost of the OEMentrant. Condition (21) implies the lower 𝑐 ∗ and/or the lower 𝐹 ∗ , the higher the profitability ofOEM entry.

4. The OEM equilibrium
3Equilibrium with OEM
Would incumbent IHM firms supply OEM products to entrants? In the standard monopolisticcompetition framework, there are still many firms, and because of that, incumbent firms take nostrategic actions, although allowing entry toughens competition. For incumbent IHM firms to gainfrom supplying to OEM firms, the unit price must be 𝑐 ∗ > 𝑐. However, if 𝑐 ∗ > 𝑐 then other IHM
firms would offer a lower 𝑐 ∗ undercutting 𝑐 ∗ . Therefore, in equilibrium,𝑐 ∗ = 𝑐.
(22)Free entry leads to zero profits both for IHM and OEM firms. Therefore, for the IHM firms,
𝑝𝑖 𝑞𝑖 − (𝐹 + 𝑐𝑞𝑖 ) + 𝑚𝑖 (𝑐 ∗ 𝑞𝑗∗ − 𝑐𝑞𝑗∗ ) = 0,(23)
where𝑛(24)∫0 𝑚𝑖 𝑑𝑖 = 𝑛∗ .
In (23) and (24), 𝑚𝑖 is the mass of OEM firms that IMS firm 𝑖 supplies to. (𝑚 would be zero for
IHM firms that do not supply OEM products.) Therefore, the third term of the left-hand-side of (x)is profits raised from supplying OEM products.
For the OEM firms, free entry leads to their profits being driven down to zero, that is,
𝑝𝑗 𝑞𝑗 ∗ − (𝐹 ∗ + 𝑐 ∗ 𝑞𝑗 ∗ ) = 0.

(25)

Next, labor market equilibrium implies
𝑛𝐹 + 𝑛∗ 𝐹 ∗ + 𝑛𝑐𝑞 + 𝑛∗ 𝑐𝑞 ∗ = 𝐿,
(26)∗∗
where 𝑛 𝑐𝑞 is total labor used for OEM production. Finally, equilibrium in the goods market
requires𝑞 = 𝑝−𝜎 𝐺 𝜎−1 𝑌(27)and𝑞 ∗ = 𝑝−𝜎 𝐺 𝜎−1 𝑌,(28)
for the varieties supplied by IHM and OEM firms, respectively.
Zero profits imply
𝑌 = 𝐿.

(29)

From (27) and (28),𝑞 = 𝑞∗ .
(30)
Substituting (19), (22) and (30) into (23) and (25),𝐹 = 𝐹∗(31)
and𝑐
𝑞 = 𝐹.
(32)
𝜎−1
Substituting (29) and (30) into (26), and using (31) we have
𝐿𝑛 + 𝑛∗ = ,(33)
𝐹𝜎which is the total mass or firms or varieties, and it is the same as in the benchmark case in which
OEM is ruled out.
Maximum firm size
Size of IHM firms is maximized when a single IHM firm supplies to all the rest of OEM firms.
The output of this IHM firm is4

𝑞=𝐹(𝜎−1)
𝑐+ 𝑛∗ 𝑞 ∗ =𝐹(𝜎−1)𝑐(𝑛∗ + 1) =(𝜎−1)(𝐿+𝐹𝜎)
𝜎𝑐.

(34)

Stability of the OEM equilibria
Once a firm operates as an OEM firm, there are no gains to be made by manufacturing on its own.
That is, no profit-seeking OEM firm attempts to switch to become an IHM firm. Therefore, theOEM equilibria is stable.
Interpretation
If 𝐹 ∗ is sufficiently lower than 𝐹 to satisfy the OEM entry condition given by (21), firms would
enter in the form of OEM. However, results show that in order to explain the coexistence of IHM
and OEM firms in the same industry, 𝐹 ∗ must rise to 𝐹 in equilibrium. This suggests that OEM
firms eventually pay more fixed costs, for example, for branding, compared to what they expected
at the time of entry.

n*𝐿⁄𝜎OEM equilibria

no OEM; benchmark0𝐿⁄
𝜎nConcluding comments
In this model, firms in the same industry become heterogeneous because of OEM and not becauseof ex-ante productivity differences; OEM makes firms lumpy, that is, makes the firms
heterogeneous in size.

5This analysis, however, only explains a particular type of OEM. The OEM product studied here isa product produced by in-house manufacturing (IHM) firms on behalf of OEM firms to be soldunder the brand name of the OEM firm. Many of the sport bicycles sold under numerous different
brand names are in fact produced by a few Taiwanese manufacturers (in the form of OEM) . Theanalysis probably applies to products in which branding is important.
It does not explain contract manufacturers that specialize solely on producing OEM products, andthe strategic interactions between contract manufacturers and OEM firms, as studied by Arruñada
and Vázquez (2006) in the business literature.

References
Arruñada, Benito and Xosé H. Vázquez. 2006. “When Your Contract Manufacturer Becomes Your
Competitor” Harvard Business Review, September issue.
Dixit, Avinash K. and Joseph E. Stiglitz. 1977 “Monopolistic competition and optimum productdiversity” American Economic Review 67, 297-308.

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