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Multiply Product Firms

And economies of scale and scope.

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Review

When we had a single product firm, cost just depended on the output of the one good. The scale economy index was seen as

S = AC/MC. Now, by definition AC = TC/Q, where Q is the level of output.

The index can be rewritten

S = (TC/Q)/MC = (TC/Q)(1/MC) = TC/QMC.

We saw this had some significance because firms with economies of scale, S > 1, had average cost falling over a range of output increases and this means there is a greater chance the industry will be concentrated with few firms. The logic is that existing firms could add addition units of output cheaper than a new firm could add units.

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Multiproduct firms

Say we have a firm that produces more than one type of product. Is there a similar measure that will give a number that expresses the economies of scale in the production of all the items the firm produces? Yes, the multiproduct scale economy index is

TC/(MC1Q1 + MC2Q2 + … + MCnQn), where the firm has n products.

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Example: Say we have a two product firm with TC = 25Q1 + 30Q2 – (3/2)Q1Q2.

If the firm made one unit of each good the TC would be

TC = 25(1) + 30(1) – (3/2)(1)(1) = 53.5.

Now MCi is the marginal cost of increasing the output of good i, holding the output of good j constant.

For the example, note MC1 = 25 – (3/2)Q2. Look at the TC function. 25Q1 has Q1 in it so in MC1 just put the 25. 30Q2 has no Q1 term so it is not in MC1. -(3/2)Q1Q2 has a Q1 term in it so in MC1 just put –(3/2)Q2. (I just followed the rules of a partial derivative.)

MC2 = 30 – (3/2)Q1.

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The index of scale economies in this example is

{25Q1 + 30Q2 – (3/2)Q1Q2} /{(25 – (3/2)Q2)Q1 + (30 – (3/2)Q1)Q2}

= {25Q1 + 30Q2 – (3/2)Q1Q2}/ {25Q1 + 30Q2 – (3/2)Q1Q2} – (3/2)Q1Q2

= TC / (TC – (3/2)Q1Q2) in this example.In this example S >1 all the time. You know B/B = 1 and if you subtract a positive number from the denominator the whole fraction becomes larger than 1.

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In the multiproduct context

If S > 1 there are global economies of scale,

S < 1 there are global diseconomies of scale, and

S = 1 there are global constant returns to scale.

Economies of Scope.

This is when you buy Scope in a really big bottle.

Economies of scope are present whenever it is less costly to produce a set of different goods in one firm than it is to produce that set in two or more firms.

Example of two goods in the set. Add up the cost of good 1 in one firm and the cost of good 2 in another firm then subtract the cost from making both in one firm.

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In notation form we have

TC(Q1, 0) + TC(0, Q2) – TC(Q1, Q2).

If this expression > 0 we have economies of scope,

If < 0 we have diseconomies of scope, and

If = 0 we do not have economies of scope.

The degree of scope economies is

Sc = {TC(Q1, 0) + TC(0, Q2) – TC(Q1, Q2)} / TC(Q1, Q2).

Why do scope economies exist?

1) Particular outputs share common inputs. Example: advertising diet coke with lemon twist also helps classic coke because the brand names are the same.

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2) Cost complementarities – when producing one good it lowers the cost of producing the other good. Having phone company in local market makes it cheaper to be in long distance as well. The infrastructure is there.

Let’s do problem 4.3 page 73 together.

TC(Q1, 0) = 2 + sqrtQ1.

Note when you have a term like sqrtQ1 in a TC the MC has a term 1/(2sqrtQ1).

Scale economy index = 2 + sqrtQ1 / (Q1{1/(2sqrtQ1})

= 2 + sqrtQ1 / (sqrtQ1/2)

=(2 + sqrtQ1) (2)/ sqrtQ1

= (4 + 2sqrtQ1) / sqrtQ1 = (4/sqrtQ1) + 2. This is always greater than 1. So scale economies exist.

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TC(0, Q2) = 2 + 2Q22.

Scale economy index = 2 + 2Q22 / {Q2(4Q2)

= (2 + 2Q22) / 4Q22

= 1/(2Q22) + 1/2

This is less than 1 for q values greater than 1 so we have diseconomies of scale.

These indexes suggests the industry with good 1 has few firms and good 2 could support many firms.

Economies of scope idea

(2 + sqrtQ1) + (2 + Q22) – (3 + sqrtQ1 + Q22) = 1 always, so there are economies of scope. Marketing and distribution channels would be shared and thus the good 2 market may be more concentrated than suggested by S.

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Network Externalities

Economies of scale and scope are not the only reasons for concentration to be high in an industry. Another reason is network externalities - a situation where a consumer’s willingness to pay for a good or service rises as the number of consumers buying the product rises.

Fax machines are an example of this.