4.4. Vertical Differentiation Vertical Differentiation

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1 Mtilde Mchdo Industril Orgniztion- Mtilde Mchdo Verticl Differentition 4.4. Verticl Differentition The Hotelling model studies situtions of horizontl differentition since for equl prices there re lwys consumers tht prefer to nd others to. Let s modify the Hotelling model to incorporte qulity differences between the goods (i.e. verticl differentition) Industril Orgniztion- Mtilde Mchdo Verticl Differentition

2 Firms nd consumers re loctes in the intervl [0,]. ll consumers prefer good close to. Consumers re uniformly distributed long [0,]. firms nd locted in nd b respectively. W.lo.g 0 b 0 b Industril Orgniztion- Mtilde Mchdo Verticl Differentition Verticl Differentition Utility of consumer x is: Consumer s loction in [0,] x p i Ux() i bx p i firm For given position x, the gross consumer surplus of buying from is higher (bx>x) so willing to py higher price. Notice tht s x increses the consumer hs higher vlution for both goods U/ x>0 Two-stge gme: st stge: Firms select their loctions (i.e. their product qulity) nd stge: Firms compete in prices simultneously) Industril Orgniztion- Mtilde Mchdo Verticl Differentition 4

3 We solve the gme bckwrds. nd stge: Suppose there is n indifferent consumer between nd : ˆx U ˆ( ) ˆ ˆ x x p bx p Ux ˆ( ) p p xˆ b Consumers to the right of left buy from : If prices re equl ll consumers buy from (i.e. the indifferent consumer is locted t zero) ˆx buy from nd to the Industril Orgniztion- Mtilde Mchdo Verticl Differentition Verticl Differentition Therefore demnd for is xˆ nd for is ˆx () () () () ecuse b> (b is higher qulity), we must hve P >P. 0 -p -p z Demnd for b ˆx Demnd for x If we require U 0 for the consumer to buy, then the demnd for is only [z, ˆx ] tht is ˆx -z, where zp / Industril Orgniztion- Mtilde Mchdo Verticl Differentition 6 3

4 Note tht if p >p ll consumers would buy from, (higher qulity nd lower prices), the indifference curves would not cross becuse there would not be n indifferent consumer. () () -p -p Industril Orgniztion- Mtilde Mchdo Verticl Differentition Verticl Differentition Let s suppose c0, the problem of firm is: p p Mxπ (,,, ) ˆ b p p px p p b π p p FOC: 0 p 0 p b b p p p 0 p for b b b Firm s rection function Industril Orgniztion- Mtilde Mchdo Verticl Differentition 8 4

5 El problem de l empres es: p p Mxπ (,,, ) ( ˆ b p p p x) p p b π p p FOC: 0 p 0 p b b p p b + p + p 0 b b b b b + p p for b Firm s rection function Industril Orgniztion- Mtilde Mchdo Verticl Differentition Verticl Differentition The equilibrium is the solution of the system: p p p b + p b + p 3 b ( b ) b p p >0 nd p > 0 MC The firm with the highest qulity chrges higher price but both firms chrge bove mrginl cost. The higher is the difference in qulity (i.e. the higher is the distnce (b-)) the higher re both prices. Industril Orgniztion- Mtilde Mchdo Verticl Differentition 0 5

6 Demnds in the nd stge re: (, ) (, ) ˆ p b p b D(, b) x b 3 p(, b) p(, b) D (, ) ( ˆ b x) b 3 Industril Orgniztion- Mtilde Mchdo Verticl Differentition 4.4. Verticl Differentition Profits re: b (, ) ˆ p p π b px p p 3 b 3 ( b ) p p π (, ) ( ˆ b p x) 3 b ( ) ( b ) b b π ( b, ) ( b ) π( b, ) (( b ) p + p) > π ( b, ) 3 9 Industril Orgniztion- Mtilde Mchdo Verticl Differentition 6

7 st stge: b Mxπ (, b) 9 π * FOC: < ( b ) Mxπ (, b) b 9 π 4 * FOC: > 0 b b 9 Principle of mximum differentition. Intuition: With verticl differentition, firms specilize in given qulity niche (high vlution consumers nd low vlution consumers). The higher is the difference in qulity the higher is the mrket power of ech firm Industril Orgniztion- Mtilde Mchdo Verticl Differentition Verticl Differentition * * ecuse 0, b, π p xˆ * * * * * 4, π 9 9, p Conclusion: Firms look for mximum differentition from their rivls. lthough qulities here hve the sme cost (c0), still firm prefers to produce n inferior good in order to differentite from the rivl. If firms choose to locte sequentilly, then the first one to enter would select b where profits re higher. Industril Orgniztion- Mtilde Mchdo Verticl Differentition 4 7

8 If consumers my choose not to buy () () 0 -p -p z ˆx Demnd for b () Demnd for () Industril Orgniztion- Mtilde Mchdo Verticl Differentition 5 x Consumers between 0 nd zp / would be better off if they do not buy. The indifferent consumer is the sme s before. ˆx p p p D( p, p) x z b p p D( p, p) x b 8