Notes on the Mussa-Rosen duopoly model Stephen Martin Faculty of Economics & Econometrics University of msterdam Roetersstraat 08 W msterdam The Netherlands March 000 Contents Demand. Firm............................... 3. Firm.............................. 6 Firm s best response function 8 3 Firm s best response function 6 4 Equiliium values 4
Demand In the most general version of the model, n consumers uniformly distributed over [ ; ], where 0. Without loss of generality, normalize n =. Firm produces a variety of quality, rm produces a variety of quality ; without loss of generality, let. Individual utility function: Quality-adjusted prices: u(; ) = p: () = p () = p (3) Consumers with get positive net utility buying from ; consumers with get positive net utility buying from ; in both cases, if the indicated intervals exist. Marginal utility of quality of the consumer who receives the same utility buying from or : p = p Note that p = p = : (4) = ( ) + ( ) and ence = + ( ) = ( ) () (6) =) (7) =) (8) is the marginal utility of quality of the consumer who gets the same net utility buying from or ; nothing requires that utility to be positive.
. Firm () When would rm supply the entire market? p p 8 and p 0 8 p ( ) p 8 and p 8 p min [p ( ); ] min ; () When would rm sell nothing? (9) ( ) ( ) or equivalently p p ; =) q a = 0; =) q a = 0; hence min ; ; =) q a = 0: (0) Normally we would automatically limit attention to 0, so we would write min ; ; =) q a = 0 () (3) now consider =) and or equivalently p and p p ; =), q = N [min( ; ) ] : () 3
Together, the conditions are or p p (3a) When is? p p p p ( ) p ( ) p Since we already require p p, the overall condition for and is q = N ( ) max [ ; p ( )] p p (3) Expressed in terms of quality-adjusted prices max ; p max ; (3b) p p ( ) =) Since we already require p p p : (4) p, the overall condition is () p min p ; p ( ) or, expressed in terms of quality-adjusted prices, min ; ; (6) and for p / in this range, When is q = :: (7) 4
so condition reduces to : (8) h (4) Now consider,, max 0; =) (as before); sells from to min ( ; ), q = We know that for [min ( ; ) ] : (9) p ( ) p or, in terms of quality-adjusted prices, ; but the latter is satis ed by assumption in the region under consideration; hence q = : (0) Summary: i 8 >< q = >: 0 ; + 0 ; + ; ()
. Firm When would rm supply the entire market? p p 8 and p 0 8 or ( ) + p p 8 and p 8 ( ) + p p and p + p p and p + and The condition for rm to supply the entire market is min ; + () nd if q = in this region, q = 0. Note: another way to interpret the condition + is as being derived from Consider the line p p p p + ( ) = + If = 0, = ence the line = and ( ; ).. If =, =. + connects the points 0; 6
When would rm sell nothing? p 0 8 q also equals zero if : p 8 p 8 p (3) p p p p p p + ( ) p p + + (4) s before, the line = connects the points 0; and ( ; ) (now thinking in terms of ( ; )-space). What about? =).; then q = ( max( ; )) = () What about, +? =) ; by its derivation, for +, ; hence q = N ( ) (6) 7
Final region: 0. y its derivation,. ence also + +, + =) ; so q = : (7) Firm s best response function Firm s payo must analyzed for two ranges of, 0 and. () 0. The quantity demanded of rm is q 8 >< >: 0 + + + (8) 0 + (.) q =, = p q = q =. The local maximum of s payo function occurs when makes as large as possible consistent with being within the limits that de ne region., that is, = + : (9) (.) q =, = p q = = The global maximum of the region.. payo function occurs at and for de ned by this rst-order condition 0; (30) = ( ) ( ) : (3) : 8
Solve the rst-order condition for = + : (3) lies within region.. if + + : (33) If this inequality is satis ed, the global maximum of the.. payo function lies within region.. and, because the.. payo function is a parabola, is rm s best response price. The right-hand inequality is always satis ed. The condition for the lefthand inequality to be satis ed is + + + + + ( ) (34) ( ) (3) Remark : if <, the right-hand side of (3) is negative and the global maximum of the. region payo function lies in region.; the local maximum of the. region payo function in region. occurs at the boundary between region. and region., and this is rm s best response price: ( ) = + : (36) Remark : we are considering the case 0. If ( ) ; (37) 9
then the left-hand inequality in (33) is always satis ed for case, the global maximum of the region. payo function lies in region. and is, by the shape of the payo function, rm s best response price. (37) can be rewritten : (38) (38) is satis ed if = and = are both su ciently large if qualities are su ciently di erent and preferences (marginal utilities of income) su ciently diverse. If then (34) is satis ed for 0 0 ; (39) ( ) ; (40) and over this range rm s best-response price is (3) ( ) = + ; while for ( ) ; (4) (34) is violated and rm s best response price is given by (36) ( ) = + : (.3) q = 0, = 0; s best response price never lies in this range. For = =, = =, = = > h i For these parameter values = + 0 =) ( ) = + = + 0 : 0
(). The quantity demanded of rm is 8 0 >< q + >: 0 + : (4) (.) 0 q =, = p q = q =. The local maximum of s payo function occurs when makes as large as possible consistent with being within the limits that de ne region., that is, = : (43) is For = =, = =, the local maximum payo on region. = q = ()() = : (.) q =, = p q = = = ( ) ( ). is a parabola with value 0 at = 0; and global maximum of the region.. payo function occurs at = ; (44) and for de ned by this rst-order condition rm s payo is (44) lies in the range that de nes region. if = 4 : (4) : (46) If ; (47)
the left-hand inequality in (46) fails, the global maximum of the region. payo function lies in region. and the local maximum of the region. payo function on region. is the boundary between region. and region., =, (43). If, the local maximum of the region. payo function on region. is = (48) with corresponding payo = 4 (49) For = =, = =, local maxima and local maximum payo s on region. are = = 4 4 () = 4 : (.3) + q =, = p q = = The global maximum of the region..3 payo function occurs at and for de ned by this rst-order condition 0; (0) = Solve the rst-order condition for = ( ) ( ) : () + lies within region..3 if + + : () (3) :
+ + : (4) The right-hand inequality is always satis ed. The left-hand inequality is satis ed for + We now consider the case. If ; () the left-hand inequality in (4) is never satis ed. () can be rewritten (6) (7) (Compare with (38) and (39)). If = and = satisfy (7), the global maximum of the region.3 payo function occurs in region. and the local maximum of the.3 payo function on region.3 is the boundary between region.3 and region., = : (8) 3
If then and est response prices and payo s are 8 h >< + = >: = 8 >< = >: + ; (9) : (60) i (6) h + 4 ( )( ) 4 ( ) ( ) = + i (6) respectively. For = =, = =, the local maximum best response function and payo on region.3 are = = ( ( 4 + () = + 0 9 0 () = 0 9 4 + (4)(4) = + 6 0 0 = 9 9 = 0 4 ( ) 9 (.4) + q = 0, = 0; s best response price never lies in this range. Summarize results for the four ranges that make up case : (.) 0 : = =, = () =. 4
(.) : = 4 = ( ) = = 6 64 64 : (.3) 4 + + = 0 0 9 0 9 ( 4 + 4 (4)(4) = = + 6 0 0 = 9 9 = 0 4 ( ) 9 (.4) 4 + : q = 0, = 0. Comparing regions. and.3, we can assemble the table..3..3 + 9 0 + 6 0 9 4 ( ) 64 4 ( ) Compare payo s at the two local maxima in each of these three regions. 9 : s payo in region. at = is. s varies from to, s payo in region.3 increases from + () 6 0 = 44 = 6 7 to + () 64 64 6 0 = 6 = 9 49 and is always greater than s 64 64 payo in region. at =. For, s best-response 9 quality-adjusted price is ( ) = + 0. 9 : s payo in region. at = is. 4 ( ) 4 + 4 0
( ) ( 4) 0 This fails for < 4, is satis ed for 4. Since we consider the range 9, the global maximum of s payo is 4 ( ) = 4 ( ) for the best-response quality-adjusted price ( ) =. : s payo in region. at = : is =64. s payo in region.3 at = is 4 ( ). This takes its maximum value 4 = for 6 = = and falls to 0 for =. The global maximum of s payo is at ( ) = :. Firm s best-response function is 8 >< ( ) = >: + 0 0 + 0 0 9 0 9 3 Firm s best response function Firm s payo must be analyzed over the following ranges: () 0 () (3) (4) + () + () 0 q = 0, = 0. Firm s payo is 0 no matter what price it sets. () (.) 0. The right-hand boundary is the line along which =. q =, =. = : (63) 6
The rst-order condition is 0; (64) from which rm s payo along the rst-order condition is = ( ) ( ) ( ) : (6) Solve (64) for = ( ) = ( ) = : (66) given by (66) lies in region. if 0 oth inequalities in (67) are satis ed for 0 : (67) ; (68) a condition which is met by the de nition of region. ence the global maximum of rm s region. payo function lies in region., and the local maximum of rm s payo function in region. is given by (66). (.) : q = 0, = 0. Firm s payo is 0 no matter what price it sets. ence rm s bestresponse quality-adjusted price in region is given by (66): ( ) = 7 for : (69)
For = =, = =, (69) is ( ) = () for () : ( ) = for 4 : This is the equation of the straight line connecting (0; 4) and ( ; ) in ( ; )-space. (3) (3.) 0 : q =, =. = = : (70) From the discussion of case., the global maximum of rm s region 3. payo function occurs at (66) = : This lies within region 3. for 0 : (7) The condition for the left-hand inequality to be satis ed is 0 ; (7) which is met by de nition of region 3. The condition for the right-hand inequality to be satis ed is 8
If + + + : (73) Case 3 is de ned for values of in the range : + ; (74) then (73) is satis ed for all of case 3., the global maximum of the region 3. pro t function occurs in region 3., and is given by (66). (74) can be rewritten + + + + + + : (7) This will be satis ed if = is su ciently large (marginal utilities of quality slightly dispersed) and = su ciently small (qualities relatively close together). If (7) is not met, then + (76) 9
For +, the right-hand inequality in (7) is satis ed, the global maximum of the region 3. pro t function occurs in region 3., and is given by (66). For +, the right-hand inequality in (7) is violated and the global maximum of the region 3. pro t lies to the right of region 3.. The local maximum of the region 3. pro t function in region 3. is the boundary between region 3. and 3., the line =. The local maximum of the region 3. pro t function in region 3. occurs at 8 < = : If = h i + + +, the value of along the rst segment is + = : : Payo s are 8 < 4 ( = [ )( ) ( ) ] + : ( ) + (78) For = =, = =, (77) is (77) = () = 6 6 4 : The rst segment connects the points ( ; ) and (; 6 ) in ( ; )-space. The second segment connects the points (; 6 ) and (; 4) in ( ; )-space. Payo s are = [ 4 ( )( ) ( ) ] 6 ( 6 6 ) 4 (3.) : q =, = 0
Rewrite the expression for as = = = ( ) (79) ( ) ( ) ( ) : (80) The rst-order condition to maximize (80) is 0; (8) from which rm s payo on the rst-order condition is = ( ) ( ) : (8) Solve (8) for = : (83) = lies within region 3. for (84) The right-hand inequality is always satis ed. The left-hand inequality is satis ed for : (8) Case 3 is de ned by the inequalities. If ; (86) then (8) is violated for all in Case 3., the global maximum of the region 3. payo function lies in region 3., and the local maximum of the 3. payo function on region 3. is the boundary between regions 3. and 3., =. (86) can be rewritten
+ If (87) is violated, then = : (87) and ( (88) For = =, = =, (88) is = 4 : The rst segment is a line connecting (; ) and (; ) in ( ; )-space. The second segment is a line connecting (; ) and (; 4) in ( ; )-space. (3.3) : q = 0, = 0. Collect results for Case 3: 3.: with payo s 8 < = : (3.) with payo s = () ( )( ) = 6 6 4 : = () ()() = 4 : ( () ()() ()() ( )( ) ()() ( ) ( ) 6 6 = ( 6 6 ) 4 : = ( 6 ) = 64 4 6: compare payo s at the two local maxima, versus ( 6 ) 6
6 6 ( ) = 6 0: 64 ence setting = gives rm the greatest payo for 6 ; ( ) = for 6. This is a straight line connecting ( ; ) and (; 6 ) in ( ; )-space. 6 : ( ) = : This is a straight line connecting (; 6 ) and (; ) in ( ; )-space. 4: compare payo s at the two local maxima, ( 6 ) ence versus 64. 64 6 ( ) = 64 ( ) > 0: ( ) = for 4: This is a straight line connecting (; ) and (; 4) in ( ; )-space. Firm s best response function in region 3 is 8 < 6 6 ( ) = : 4 (4) + (4.) 0 : q =, = The right boundary is the line along which =. maximizes its pro t in this region by making as large as possible consistent with remaining in the region, = ; (89) 3
and s payo is = = ( ) : (90) For = =, = =, (4.) = ( 4), = ( 4) : : q =, = = The rst-order condition to maximize (9) is : (9) 0; (9) from which rm s pro t along the rst-order condition is = ( ) ( )( ) : (93) Solve (9) for = : (94) de ned by (94) lies within region 4. for The left-hand inequality is satis ed for ( ) : 4 : (9)
Case 4 is de ned by the inequalities If + ( ) : (96) ( ) + : (97) ( ) ; (98) ; (99) then (96) and the left-hand inequality in (9) are satis ed for all in case 4; the global maximum of the region 4. payo function does not lie in region 4.. If (99) is not satis ed, then ( ) + and for ( ) + (00) the left-hand inequality in (9) is violated; the global maximum of the region 4. payo function lies in region 4., and the local maximum of the region 4. payo function on region 4. is at the boundary between region 4. and 4.. The right-hand inequality in (9) is satis ed for +
From (97), if + + (0) + + ; (0) then (0) is satis ed for all in case 4 and the global maximum of the region 4. payo function does not lie to the right of region 4.. Rewrite (0) as If + : (03) min ; + (04) then (99) and (03) are both satis ed and the global maximum of the region 4. payo function lies within region 4. for all in region 4. If (04), = If for + ; (0) then the right-hand inequality is satis ed for all in region 4, the lefthand inequality is not satis ed for some or all in region 4. Noting that we have for this case ( ); (06) + : 6
( ) + + (07) If (07), both inequalities in (9) are satis ed in the lower range of, ( ), while the left-hand inequal- ( ) ity is violated for the upper range of, +. 8 < : h If = i ( ) ( ) + (08) + ; (09) then the left-hand inequality in (9) is satis ed for all in region 4, + ( ) the right-hand inequality is violated for some in region 4, + + : ence if (09) + + ( ): (0) There are two subcases to consider, + + ( ): () 7
and + The condition for () is + + ( ): () If (3), then + + + + + + : (3) 8 < : h with payo s 8 < : = i + ; + + ( ( ) )) 4( )( ) ( )( ) ( ) If in contrast = + (4) + : + () + + ; (6) 8
then in + + ( ); + ( ) implies that the left-hand inequality in (9) is satis ed for all in case 4, while + + implies that the righthand inequality is violated for all in case 4. The global maximum of the region 4. payo function lies to the right of region 4. throughout, and the local maximum of the region 4. payo function on region 4. is the boundary between region 4. and region 4.3, = for with payo = ( ) ( ) ( ) for + (7) (8) For = =, = =, region 4 is de ned by the inequalities () + () 4 4 : + : () is satis ed + + ( ) 4 4 7 ; = for 4 4 with payo = 6 ( ) for 4 4 : 9
(4.3) : q =, = =. ( ) ( ) ( ) : (9) From (8), rm s payo along the rst-order condition is = ( ) ( ) ; and the global maximum of (9) occurs at (83) = : de ned by (83) lies within region 4.3 for : (0) The right-hand inequality is always satis ed. The left-hand inequality is satis ed for Region 4 is de ned by the inequalities If + ; () then the left-hand inequality in (0) is satis ed for all in case 4. Rewrite () as + : () If () is satis ed, then so is the left-hand inequality in (0), and the global maximum of the region 4.3 payo function lies in region 4.3. 30
If () is not satis ed, then ; for min ; + (3) the local maximum of the region 4.3 payo function on region 4.3 is the boundary between region 4. and 4.3. For = =, = =, region 4 is de ned by the inequalities () is satis ed: 4 4 : + : The global maximum of the payo function lies in region 4.3 for ; and this is satis ed for in the range 4 4. s payo is ()() = ( ) ( ) = 64 (4.4) : q = 0; = 0. Firm s best-response price would never be found in this region. Collect results for case 4, 4 4. (4.) 0 ( 4): = ( 4), = ( 4) : (4.) 0 : = =, = 6 ( ) (4.3) ; = ; = 64 The region 4. payo is never less than the region 4. payo : 6 ( ) ( 4) = 6 (63 ): 3
This di erence is positive for = 4 and declines to zero for = 4. For these parameters, the global optimum never occurs in region 4.. Now compare the region 4.3 payo and the region 4. payo : 64 6 ( ) = 64 ( ) ; which is positive. earns the greatest pro t at the local maximum of its payo function on region 4.3. s best-response quality-adjusted price is therefore ( ) = for 4 4 : This is a straight line connecting (; 4) and (:; 4:) in ( ; )-space. () + (.) 0 : q =, = Firm maximizes its payo charging the highest price consistent with being in region., =. s payo is =. (.) : q =, = The right boundary is the line along which =. = The rst-order condition to maximize (4) is ( ) : (4) 0; () from which s payo along the rst-order condition is = : (6) Solve () for = : (7) = lies in region. for : (8) 3
The left-hand inequality is satis ed for : (9) If (9) is satis ed, the global maximum of the region. payo function does not lie in region.. If (9) is violated, the global maximum of the region. payo function lies in region., and the local maximum of the region. payo function on region. is the boundary between regions. and.. The right-hand inequality in (8) is satis ed for + : (30) Region is de ned by the inequalities + : If + ; (3) then (30) and the right-hand inequality in (8) are satis ed for all in case.; the global maximum of the region. payo function does not lie to the right of region.. Rewrite (3) as 33
: (3) (3) is the contrary of (9). Except in the borderline case =, if the left-hand inequality in (8) is satis ed, the right-hand inequality is not, and vice versa. If (3) is satis ed, the global maximum of rm s region. payo function does not lie to the right of region.. If (3) is satis ed, (9) is violated; the global maximum of rm s region. payo function lies in region. and the local maximum of rm s region. payo function is the boundary between region. and region., =. If (3) is violated, then For + + : (33) ; (34) (30) and the right-hand inequality in (8) are violated. The local maximum of rm s region. payo function on region. is the boundary between region. and.3, Firm s payo is For = = : (3) ( ) ( ) + = ( ) = (36) ( ) (37) ; (38) (30) and the right-hand inequality in (8) are satis ed. The global maximum of rm s region. payo function lies within region.. 34
ence if (3) is violated 8 < = : with payo s 8 < = : ( ) ( ) 4 (39) (40) For = =, = =, the global maximum of s region. payo function is = = : Region. is de ned by the inequalities = = for in the range. The best response and payo functions are 0 = ; () 0 =. = ( 4) and = ( 4) 4 4 4 = ( 4 ) ( 4) 4 4 4 6 :: respectively. (.3) : q =, = s in the discussion of region (4.3), = ( ) ( ) ( ) : (4) The rst-order condition to maximize (4) is 0; (4) 3
from which rm s pro t along the rst-order condition is = ( ) ( ) : (43) Solve (??) for = : (44) de ned by (44) lies within region.3 for : (4) The right-hand inequality is always satis ed. The left-hand inequality is satis ed for (46) Region is de ned by the inequalities + : The right-hand side of (47) is less than : =! + =! = (47) > 0: (48) 36
If + (49) the left-hand inequality in (46) is violated for all in. Rewrite (49) as + + + : (0) If (0) is satis ed, the left-hand inequality in (46) is violated for all in region and the local maximum of the region.3 payo function on region.3 is the boundary between region. and region.3, = 37 ()
with payo = = ( ) ( ) ( ) ( ) If (0) is violated, then For = + + + ( ) ( ) ( ) : () : (3) ; (4) the left-hand inequality in (4) is satis ed, the local maximum of the region.3 payo function lies in region.3, s best response price is (44) with payo (43). For () the left-hand inequality in (4) is violated, the local maximum of the region.3 payo function lies to the left of region.3, s payo -maximizing price on region.3 is the boundary between region.3 and region., = with payo (). Overall, if (0) is violated, ; (6) 8 >< >: = + (7) 38
8 >< >: 4 ( )( ) = h ( ) i = + (8) For = =, = =, region.3 is de ned by the inequalities ( 4) ; with in the range 4 and the global maximum of s region.3 payo function at = : For the global maximum of the payo function to lie within the region, we must have 0 0 40 9 40 40 9 = 44 9 : The local maxima and payo functions are = = 4 4 4 9 ( 4) 4 4 9 64 4 4 4 9 ( 4) 4 4 9 (.4) : q = 0, = 0. Firm s best-response price would never be found in this region. Collect results for case, 4.. 0 : =, =. ( 4) = ( 4) 4 4 4 39
and = ( 4 ) ( 4) 4 4 4 6 ::.3 ( 4) = 4 4 4 9 ( 4) 4 4 9 = 64 4 4 4 9 ( 4) 4 4 9 Compare the best payo in region. with payo s in region.: On the range 4 4 : 4 ( ) ( 4) = 4 0 4 On the range 4 : 6 > 0: Firm s best response price is never in region.. Compare payo s in regions. and.3: 4 4 4 9 4 ( ) ( 4) = 64 8 40 9 64 0: The payo in region.3 is never less than the payo in region., and in general greater: ( ) = : This is a straight line connecting (:; 4:) and ( 9 ; 4 4 9 ) in ( ; )- space. 4 4 9 4 : 40
For 4 4 9 4, 4 ( ) ( 4) ( 4) = ( 4) 4 ( ) = 4 ( 4) ( ) < 0 ( ) = ( 4) : This is a straight line connecting ( 9 ; 4 4 9 ) and ( ; 4 ) in ( ; )-space. 4 6 = ( 4) 4 6 ; which is negative on the range considered. For 4, ( ) = ( 4) : This is a straight line connecting ( ; 4 ) and (; ) in ( ; )-space. Summarize results for case : ( ) = 4 4 4 ( ; 4 ) ( ; 4 4) 9 0 9 9 ( 4) 4 4 9 ( ; 4 4 ) (; ) 9 9 Summarize results for the numerical example for all regions: : ( ) = for 4 : This is a straight line connecting (0; 4) and ( ; ) in ( ; )-space. 3: 8 < 6 ( ; ) (; 6) ( ) = : 6 (; 6 ) (; ) 4 (; ) (; 4) 4
4: ( ) = for 4 4 : This is a straight line connecting (; 4) and ( ; 4 ) in ( 0 ; )-space. : ( ) = 4 4 4 ( ; 4 ) ( ; 4 4) 9 0 9 9 ( 4) 4 4 9 ( ; 4 4 ) (; ) 9 9 Combining intervals where possible, 8 >< ( ) = >: 4 Equiliium values 4 6 (0; 4) (; 6) 6 (; 6 ) (; ) 4 4 (; ) ( ; 4 4) 9 9 9 ( 4) 4 4 9 ( ; 4 4) ( ; 4 ) 9 9 The two reaction functions intersect on the segments ( ) = = 4 ( ) ( ) ( ) = + = 4 ( ) ( ) or, for the numerical example, + ( ) = ( ) = + 0 = 64 ( ) = 6 Solve the equations of the reaction functions: + 0 = = + 4
= + = + = = = 4 = 4 : Evaluate these for the numerical example: Solve the numerical versions: = 4 = 0 9 : () = 40 9 = + 0 = + 0 9 0 = = 40 9 = 9 = 0 9 = 9 In general terms, payo s are = 4 ( ) ( ) 4! = ( ) ( ) 43 4!
0 @ @ @ ( ) 4! = (4 7 ) (4 ) 3 which is positive for the values of the numerical example. = 4 ( ) ( ) " 4 = 4 ( ) ( ) = 4 ( ) ( ) = 4 ( ) ( ) " 4 4 = 4 ( ) ( ) 0 @ @ @ ( ) 4 4 + + 4 4 +!!!! = 4 3 + (4 ) 3 which is positive for the values of the numerical example. # # 44