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Existence of Natural Monopoly in Multiproduct Firms

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

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Existence of Natural Monopoly in Multiproduct Firms

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  1. ExistenceofNaturalMonopoly in MultiproductFirms Competition Policy and Market Regulation MEFI- Università di Pavia

  2. MultiproductSub-additivity • Twoproducts q1, q2 • Costfunction. C(q1, q2) • Def.: qi a vectorof the 2 products: qi = (q1i , q2i ) • N vectorssuchthat:∑i q1i=q1 and ∑i q2i=q2 • Sub-additivecostfunction: C(∑i q1i , ∑i q2i) = C (∑iqi) < ∑iC (qi)

  3. Whatdrivesmultiproductsub-additivity? • Economiesof scope: C(q1, q2)< C(q1,0)+ C(0, q2) • Multiproducteconomiesof scale • DecliningAverageCostfor a specificproduct • Decliningrayaveragecost (varyingquantitiesof a set of multiple products, bundled in fixedproportions)

  4. DecliningAverageIncrementalCost • Incrementalcostof production for q1 (holding q2constant): IC(q1I q2) = C(q1, q2) - C(0, q2) • Averageincrementalcost: AIC =[C(q1, q2) - C(0, q2)] /q1 If AIC ↓ when q1↑ :decliningaverageincrementalcostof q1 A measureof single producteconomiesof scale in a multiproductcontext We can seeif the costfunctionhasdecliningaverage IC foreachproduct

  5. Declining Ray AverageCosts • Fix the proportionof multiple products: (q1/q2= k) • Whathappenstocostsifweincreasebothproducts output holding K constant? • Does the averagecostof the bundle decreaseas the sizeof the bundle increases?

  6. Declining Ray AverageCosts • We can considerdifferentproportions k, and seeifwehaveeconomiesof scale alongeachray k in the q1, q2space • Wehavemultiproducteconomiesof scale foreachcombinationof q1/q2if: C(λ q1, λq2) < λC(q1,q2)

  7. Declining Ray AverageCosts: Examples • Consider C(q1,q2 ) = q1+ q2+ (q1q2)1/3 • Itischaracterizedbymultiproducteconomiesof 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)

  8. No Multiproductsub-additivity • HOWEVER thiscostfunctionexhibitsdiseconomiesof scopeas: 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 thiscostfunctionisnot sub-additive, despitemultiproducteconomiesof scale, aseconomiesof scope are lacking • Itis more convenientto produce the twoproducts in two separate firmsNo NaturalMonopoly

  9. An examplewithmultiproductsub-additivity • Sub-additivity in a multiproductcontextrequiresbothcostcomplementarity (economiesof scope) and multiproducteconomiesof scale, over at least some rangeof output. • Consider the followingcostfunction: C(q1,q2 ) = q11/4+ q21/4 -(q1q2)1/4 • Itexhibitseconomiesof scope (..look at -(q1q2)1/4 ) C(q1,0)+ C(0, q2) = q11/4+ q21/4 > q11/4+ q21/4 -(q1q2)1/4 = C(q1,q2 ) Then: C(q1,q2 ) < C(q1,0)+ C(0, q2)

  10. An examplewithmultiproductsub-additivity: C(q1,q2 ) = q11/4+ q21/4 -(q1q2)1/4 • Itexhibitsmultiproducteconomiesof scale (foranycombination K of the twooutputs the costof production ofthiscombinationincreaseslessthanproportionallywithanincrease in the scale of the bundle,… byvirtueofpower ¼ in the costfunction) • For the samereasonitexhibitsproductspecificeconomiesof scale (decliningaverage IC, at any output) • It can beshownitis a globallysub-additive costfunction (i.e. sub-additive at everylevelof output)

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