# Prices and Profits: Multiproduct Monopolist

A multiproduct monopolist produces two goods which may be substitutes or complements to one another.

In this topic we consider:

1. The determinants of the multiproduct monopolist’s optimal price
2. Extension of the break even analysis to cover the multiproduct monopolist.

Multiproduct monopolist’s demand

Assume the monopolist produces two goods: Koque (good 1) and Phantah (good 2).

The marginal benefit of Koque depends on how much Koque and how much Phantah consumed. Thus consumer’s willingness to pay depends on how much of each product is consumed:

We write:

P1(Q1,Q2)

and:

P2(Q1,Q2)

The firm’s revenue from the production of Koque is:

R1 = P1(Q1,Q2)Q1

and Phantah is:

R2 =P2(Q1,Q2)Q2

Total revenue is:

R = P1(Q1,Q2)Q1 + P2(Q1,Q2)Q2

Assume that total cost from production is:

C = c1 Q1+c2 Q2

Where c1 is the marginal cost of Koque and c2 is the marginal cost of Phantah.

To derive the rule for the profit maximising output of Koque, consider the impact of an increase in output by DQ1. The increase in revenue is given by:

DR1 = DP1Q1 + P1DQ1+ DP2Q2

and the increase in cost is given by:

DC = c1DQ1

Profit maximisation requires, MR1=MC1, that is:

DR1/DQ1= DC/DQ1

or:

DP1Q1/DQ1+ P1+ Q2(DP2/DQ1)  = c1

or:

P1[1 + ] = c1

or:

P1[1 + ] = c1

Now define:

ε = –

ε is the own price elasticity of demand, and:

ε2 =

ε2 is the cross price elasticity of demand. Note that for substitute products

<0

An increase in the consumption of good l lowers the willingness to pay for good 2. (Think of different flavours of ice cream.) In this case ε2 < 0.

For complementary products:

>0

An increase in the consumption of good l increases the willingness to pay for good 2. (Think of coffee and cake.) In this case ε2 > 0.

So we can write the above formula, for the profit maximising price, as:

P1=

or:

P1=

where:

The profit maximising price of Koque (good 1) falls if the “effective elasticity” εT is increased.

The “effective elasticity” εT is increased by.

1. An increase in the own price elasticity of demand
2. A reduction in the cross price elasticity of demand.
3. a reduction in good 1’s share of revenue (i.e. a fall in R1/R2)

Note the effective elasticity is:

1. greater than the own price elasticity if goods 1 and 2 are complements (ε2 >0).
2. less than the own price elasticity if goods 1 and 2 are substitutes (ε2<0).

These impacts on effective elasticity are reflected in the profit maximising price.

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