Existence of Natural Monopoly in Multiproduct Firms

Competition Policy and Market Regulation

MEFI- Università di Pavia

Multiproduct Sub-additivity

• Two products q1, q2

• Cost function. C(q1, q2)• Def.: qi a vector of the 2 products:

qi = (q1i , q2

i )

• N vectors such that:∑i q1i =q1 and ∑i q2

i =q2

• Sub-additive cost function:C(∑i q1

i , ∑i q2i ) = C (∑i qi) < ∑i C (qi)

What drives multiproduct sub-additivity?

• Economies of scope:

C(q1, q2)< C(q1,0)+ C(0, q2)

• Multiproduct economies of scale

1. Declining Average Cost for a specific product

2. Declining ray average cost (varying quantities of a set of multiple products, bundled in fixed proportions)

Declining Average Incremental Cost• Incremental cost of production for q1 (holding q2 constant):

IC(q1I q2) = C(q1, q2) - C(0, q2)

• Average incremental cost:AIC =[C(q1, q2) - C(0, q2)] /q1

If AIC ↓ when q1↑ : declining average incremental cost of q1

A measure of single product economies of scale in a multiproduct context

We can see if the cost function has declining average IC for each product

Declining Ray Average Costs

• Fix the proportion of multiple products:(q1/q2= k)

• What happens to costs if we increase both products output holding K constant?

• Does the average cost of the bundle decrease as the size of the bundle increases?

Declining Ray Average Costs

• We can consider different proportions k, and see if we have economies of scale along each ray k in the q1, q2 space

• We have multiproduct economies of scale for each combination of q1/q2 if:

C(λ q1, λq2) < λC(q1,q2)

Declining Ray Average Costs: Examples

• Consider C(q1,q2 ) = q1+ q2+ (q1q2)1/3 • It is characterized by multiproduct economies

of scale as: λC(q1,q2)= λq1+ λq2+ λ (q1q2)1/3

C(λq1, λq2) = λq1+ λq2+ λ1/3(q1q2)1/3

and C(λq1, λq2) < λC(q1,q2)

No Multiproduct sub-additivity• HOWEVER this cost function exhibits diseconomies of scope as:

C(q1,0) = q1

C(0, q2) = q2

C(q1,0)+ C(0, q2) = q1+ q2 < q1+ q2+ (q1q2)1/3 = C(q1,q2 )

• THEREFORE this cost function is not sub-additive, despite multiproduct economies of scale, as economies of scope are lacking

• It is more convenient to produce the two products in two separate firmsNo Natural Monopoly

An example with multiproduct sub-additivity

• Sub-additivity in a multiproduct context requires both cost complementarity (economies of scope) and multiproduct economies of scale, over at least some range of output.

• Consider the following cost function: C(q1,q2 ) = q1

1/4+ q21/4 -(q1q2)1/4

1. It exhibits economies of scope (..look at -(q1q2)1/4 )

C(q1,0)+ C(0, q2) = q11/4+ q2

1/4 > q11/4+ q2

1/4 -(q1q2)1/4 = C(q1,q2 )

Then: C(q1,q2 ) < C(q1,0)+ C(0, q2)

An example with multiproduct sub-additivity: C(q1,q2 ) = q1

1/4+ q21/4 -(q1q2)1/4

1. It exhibits multiproduct economies of scale (for any combination K of the two outputs the cost of production of this combination increases less than proportionally with an increase in the scale of the bundle,… by virtue of power ¼ in the cost function)

2. For the same reason it exhibits product specific economies of scale (declining average IC, at any output)

3. It can be shown it is a globally sub-additive cost function (i.e. sub-additive at every level of output)