Bertrand%20Equilibrium - Bertrand Price-setting 1 Bertrand...

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Bertrand Price-setting 1 Bertrand Equilibrium [Like the Cournot model] Firms 1 and 2 sell identical product. [Unlike the Cournot model] Firms compete by choosing the price. To simplify notation, i , we denote its rival by j [if i = 1 ; then j = 2; if i = 2 ; then j = 1 ]. Let C i ( q i ) = c i q i ; i = 1 ; 2 (1) [Of course, c i > 0 i sells the product at a price p i ; and the bu y ers buy only from the &rm o/ering the lower price; the same price, each gets half of the entire demand. Formally, let the market demand function be given by q ( p ) = A p for 0 ± p ± A: (2) Assume that 0 < c i < A for i = 1 ; 2 i is given by: q i ( p i ) = 8 < : 0 if p i > p j A p 2 if p i = p j = p A p i if p i < p j (3) i given by i ( p 1 ; p 2 ) = 8 < : 0 if p i > p j ( A p )( p c i ) 2 if p i = p j = p ( A p i ) ( p i c i ) if p i < p j (4) equilibrium in this model. A Nash equilibrium in the Bertrand model is a pair ( p ± 1 ; p ± 2 ) satisfying 1 ( p ± 1 ; p ± 2 ) ² 1 ( p 1 ; p ± 2 ) AND 2 ( p ± 1 ; p ± 2 ) ² 2 ( p ± 1 ; p 2 ) [ Interpret ] (5) 1
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Theorem 1 ( i ) If c 1 = c 2 = c , then ( c; c ) ( i:e:; p 1 = p 2 = c ) is a Nash equilibrium. ( ii ) When c 1 > c 2 we have the following: ( iia ) If c 1 [ A + c 2 ] = 2 , then ( c 1 ; [ A + c 2 ] = 2) is a Nash equilibrium; and ( iib ) If c 1 < [ A + c 2 ] = 2 ; then there is no Nash equilibrium. Proof. ( i ) We have to verify that 1 ( c; c ) 1 ( p 1 ; c ) [in other words, if Firm 2 chooses price c; c ];and, also verify that 2 ( c; c ) 2 ( c; p 2 ) : Only the ±rst inequality is veri±ed below. Note ±rst that 1 ( c; c ) = 2 ( c; ; c ) = 0 : (6) Now, 1 ( p 1 ; c ) = & 0 if p 1 > c; ( A ± p 1 ) ( p 1 ± c ) if p 1 < c (7) Hence, 1 ( p 1 ; c ) ² 0 = 1 ( c; c ) : (8) Verify the other inequality to complete the proof. ( ii ) Remember that in both ( iia ) AND ( iib ) ; c 1 > c 2 : The following mathematical result is used: Fix a c satisfying 0 < c < A: Then the quadratic function f ( p ) = ( A ± p )( p ± c ) attains its maximum (1 = 4)( A ± c ) 2 > 0 at ( A + c ) = 2 [the midpoint between
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This note was uploaded on 12/23/2009 for the course ECON 3130 at Cornell University (Engineering School).

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Bertrand%20Equilibrium - Bertrand Price-setting 1 Bertrand...

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