1. Existence of Natural Monopoly in Multiproduct Firms Competition Policy and Market Regulation MEFI- Università di Pavia

2. Multiproduct Sub-additivity Two products q 1, q 2 Cost function. C(q 1, q 2 ) Def.: q i a vector of the 2 products: q i = (q 1 i, q 2 i ) N vectors such that:∑ i q 1 i =q 1 and ∑ i q 2 i =q 2 Sub-additive cost function: C(∑ i q 1 i, ∑ i q 2 i ) = C (∑ i q i ) < ∑ i C (q i )

3. What drives multiproduct sub- additivity? Economies of scope: C(q 1, q 2 )< C(q 1,0)+ C(0, q 2 ) 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)

4. Declining Average Incremental Cost Incremental cost of production for q 1 (holding q 2 constant): IC(q 1 I q 2 ) = C(q 1, q 2 ) - C(0, q 2 ) Average incremental cost: AIC =[C(q 1, q 2 ) - C(0, q 2 )] /q 1 If AIC ↓ when q 1 ↑ : declining average incremental cost of q 1  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

5. Declining Ray Average Costs Fix the proportion of multiple products: (q 1 /q 2 = 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?

6. Declining Ray Average Costs We can consider different proportions k, and see if we have economies of scale along each ray k in the q 1, q 2 space We have multiproduct economies of scale for each combination of q 1 /q 2 if: C(λ q 1, λq 2 ) < λC(q 1,q 2 )

7. Declining Ray Average Costs: Examples Consider C(q 1,q 2 ) = q 1 + q 2 + (q 1 q 2 ) 1/3 It is characterized by multiproduct economies of scale as: λC(q 1,q 2 )= λq 1 + λq 2 + λ (q 1 q 2 ) 1/3 C(λq 1, λq 2 ) = λq 1 + λq 2 + λ 1/3 (q 1 q 2 ) 1/3 and C(λq 1, λq 2 ) < λC(q 1,q 2 )

8. No Multiproduct sub-additivity HOWEVER this cost function exhibits diseconomies of scope as: C(q 1,0) = q 1 C(0, q 2 ) = q 2 C(q 1,0)+ C(0, q 2 ) = q 1 + q 2 < q 1 + q 2 + (q 1 q 2 ) 1/3 = C(q 1,q 2 ) 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 firms  No Natural Monopoly

9. 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(q 1,q 2 ) = q 1 1/4 + q 2 1/4 -(q 1 q 2 ) 1/4 1.It exhibits economies of scope (..look at -(q 1 q 2 ) 1/4 ) C(q 1,0)+ C(0, q 2 ) = q 1 1/4 + q 2 1/4 > q 1 1/4 + q 2 1/4 -(q 1 q 2 ) 1/4 = C(q 1,q 2 ) Then: C(q 1,q 2 ) < C(q 1,0)+ C(0, q 2 )

10. An example with multiproduct sub-additivity: C(q 1,q 2 ) = q 1 1/4 + q 2 1/4 -(q 1 q 2 ) 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)