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Multi-Product Firms

Multi-Product Monopolist

How much output should a monopolist produce and what price should they charge if the

monopolist is selling more than one product in order for the firm to maximize its profits?Well lets take the simplest case and look at a multi-product monopolist. Before we getstarted, lets suppose that the monopolist is only producing two goods (although the resultsgeneralize for more than two products) and for now that the demand for these two goodsare unrelated. By this I mean that changes in the price of one of the goods does not affectthe quantity demanded for the other good. We will explore the related goods case later on.Also, lets use our familiar linear inverse demand specification and lets suppose that themonopolist sets the price of each good uniformly. Finally, lets suppose there are no costsynergies (i.e. the cost of producing one good does not affect the cost of producing theother good) for now.

Independent Goods with No Cost SynergiesGiven that the firm is a monopolist in each separate product market and that there are nocost synergies, the profit maximizing output and price are the same as under the uniformpricing monopolist. Thus the monopolists optimal decision is the same for each product as

it was for only one product. Thus if the inverse demand is given by iiii QbaP , where

i={1,2} and the cost function is given by iiii fqeqC )( , then the optimal amount of

output for the monopolist to produce is1

11

12

*b

eaq and

2

22

22

*b

eaq . The profit

maximizing price for each good of:2

* 111

eaP and

2* 22

2

eaP .

Independent Goods with Cost Synergies

With cost synergies where the firms costs differ by producing both goods together as opposed toproducing each good separately, output and pricing decision are affected by the extent of the cost

synergies. So given: 212122112

22

2

1121 ),( qqeffqeqeqdqdqqC c , where ec

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like: 212122112

22

2

1122221111 )()( qqeffqeqeqdqdqqbaqqba c .

Skipping the algebra, the profit maximizing outputs of q1 and q2 are:

2

2211

221122*

1))((4

)())((2

c

c

edbdb

eaeeadbq and

2

2211

112211*

2))((4

)())((2

c

c

edbdb

eaeeadbq . Thus

with cost synergies, the optimal quantities for each good are more complex given the overlap incosts for each product. The greater the amount of cost synergies (i.e. the more negative ec is), thegreater the optimal amount of output for each good.So in sum, a uniform pricing monopolist producing independent goods without cost synergiesresults in the optimal amount of output exactly like under the single product uniform pricingmonopolist. If the additional characteristic of cost synergies is included, the optimal amount ofoutput is more complicated by the synergies and the greater the cost synergies in producing thetwo goods the greater the profit maximizing amount of output for each good.

Substitute Goods with no Cost Synergies

Suppose the monopolist produces two goods and these two goods are economically related,such as consumers view the two goods as substitutes or complements. Here I will go overthe substitute good case, and the next sub-section will briefly look at the complement goods

case. First, lets use thelinear demand equation for each product: 211111 pgpbaq and

122222pgpbaq . Here notice that a new term is included in the demand equation.

The parameter g is the degree that consumers view a change in the price of one product onthe demand for the other product. As g gets to zero the more consumers view each productas more differentiated. A substitute goods monopolist profit equation is:

][][][][),( 12222221111112222221111121 pgpbaepgpbaepgpbappgpbapqq

and after some tedious algebra the equilibrium price for good one is:

2

2121

1122221221112*1

)(4))(()(2

ggbbgeebagggeebabp and the equilibrium price for good

two is:2

2121

2211121112221*

2)(4

))(()(2

ggbb

geebagggeebabp . Notice as consumers view

each substitute good as being more different (i.e. less of a substitute), that the equilibriumprice increases.We notice that changes in the price of good i={1,2} has three effects: firstincreases in theprice of good i leads to increases in the firms profits due to greater additional revenue dueto the higher price; second an increase in the price of good i leads to decreases in profitsfrom lower quantity of good j; and third an increase in the price of good i increases thedemand for good j.Although not always a monopolist, think about how a Hollywood firm studio determines theprice to sell to the movie theater for a movie viewed at the theater versus a DVD or thinkabout a pharmaceutical producing both a brand name drug and a generic for the sameillness.

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Complement Goods with no Cost Synergies

If the uniform pricing multi-product monopolist produces complementary goods such as asports team selling tickets (seats) and concessions, or a satellite radio firm selling radiosubscriptions and radio players, the firm must take into account the previous three effects

when setting its optimal price. Again, using a the linear demand equation for each product:211111

pgpbaq and 122222 pgpbaq , and skipping the math, the optimal prices

for both goods are:2121

22211112

14

)()(2*

ggbb

ebagebabp and

2121

11122221

24

)()(2*

ggbb

ebagebabp . Here note that as consumers view the two

complementary goods as being more differentiated from each other, that the optimalresponse by the monopolist is to lower the equilibrium price as opposed to the substitutegoods case.