MANAGEMENT SCIENCE oi 0.287/mnsc.080.0970ec pp. ec ec56 e-companion ONLY AVAILABLE IN ELECTRONIC FORM inorms 2009 INFORMS Electronic Companion Eects o E-Waste Regulation on New Prouct Introuction by Erica Plambeck an Qiong Wang, Management Science, oi 0.287/mnsc.080.0970.
ec2 Proos, Construction o Numerical Examples in 5 an Supplement to 7 Some o our results assume a Cobb-Douglas quality unction, as e ne in (6), which satis es q(; x) = x ; q = q(; x) ; an q x = q(; x) : (EC.) x We will begin with two lemmas that are relevant to such results, then prove all propositions state in the paper, prove that the uopoly equilibrium characterize by (5) remains a sequential equilibrium when = 0, + = an we give each rm the option o introucing arbitrarily many new proucts in a row, an nally give etails on the construction o the numerical examples in 5. Lemma The monopoly equilibrium evelopment time m is a unique solution to c = q () = [ ()] () () (EC.2) an the equilibrium expeniture an quality improvement per new prouct are given by Proo: x m = [ ( m )] ( m ) ; q( m ; x m ) = [( m )] ( m ) T hus x m = q an xm = m (q + q ) : (EC.3) By Proposition (or general quality unction, prove below), the monopoly equilibrium ( m ; x m ) is the unique solution: q (; x)() q(; x) + x + c = 0 (EC.4) q x (; x)() = : (EC.5) Substituting the expression or q x in (EC.) into (EC.5), q x ( m ; x m )( m ) = q( m ; x m ) ( m ) = ( m ) x m (x m ) ( m ) = ; (EC.6) an (EC.3) ollows immeiately by solving the last equality in (EC.6) or x m an q( m ; x m ) = (x m ) ( m ), respectively. Substituting the expression or q in (EC.) an the expressions or x an q in (EC.3) into (EC.4) establishes (EC.2).
ec3 Lemma 2 The uopoly equilibrium evelopment time is the unique solution to c = q = () ( + ) ( ()) () [( + )] () ( + ) () (EC.7) an the uopoly equilibrium expeniture an quality improvement per new prouct are given by x = [ ] [( + ) ] ; q(( + ) ; x ) = [( )] [( + ) ] T hus x = q an x = @q @ + q = (( + )q + q ) (EC.8) where @q @ = ( + )q (( + ); x) = q : Proo By Proposition 4 (or general quality unction, prove below), the uopoly equilibrium is symmetric an the unique solution to: q (( + ); x)() q(( + ); x) + (x + c) = 0 (EC.9) q x (( + ); x)() = (EC.0) Substituting the expression or q x in (EC.) into (EC.0) an using q(( + ) ; x ) = (x ) (( + ) ) ; q x (( + ) ; x )( ) = (x ) [( + ) ] x ( ) = ; (EC.) an (EC.8) ollows immeiately by solving the last equality in (EC.) or x an inserting the solution into the expression o q. Using x = q an q = =(( + ) ) in (EC.9), an then substituting the expression or q rom (EC.), c = [ q (( + ) ; x )( ) + q(( + ) ; x ) x ]
ec4 = q(( + ) ; x ) = [( )] [( + ) ] ( ) ( ) ( + ) ( ) ( + ) ( ) (EC.2) which establishes is the unique solution to (EC.7): Proo o Proposition : The equilibrium ( m ; x m ) is characterize by three conitions: iscounte pro t maximization by the monopolist, stationarity, an rational expectations, expresse as ollows: ( m ; x m ) = arg maxe [q(; x)(e) x c + m ]g; (EC.3) 0;x0 m = e m [q( m ; x m )(e) x m c + m ]; (EC.4) e = m : (EC.5) The monopolist can obtain zero iscounte pro t by setting m =. Thereore, we can rule out a bounary solution with m = 0 or x m = 0, which woul result in negative iscounte pro t. Consequently, any solution to (EC.3)-(EC.5) must satisy the rst-orer conitions e q (; x) q(; x) + x + c = 0 (EC.6) q x (; x)() = : (EC.7) For any (; x) that satis es (EC.6) an (EC.7), must be a solution to H m () q (; x())() q(; x()) + x() + c () = 0 (EC.8) where x() is the unique solution to (EC.7): It ollows that lim H m() = lim [q(; x()) () x() c] :!! () By assumption, there exists some 0 such that q ( 0 ; x( 0 )) ( 0 ) x( 0 ) c > 0. Because q(; x()) () x() c increases in, the above limit is negative. Moreover lim H m() lim!0 () > 0,!0 c
ec5 so there exists some m such that H m ( m ) = 0. To show m is unique, we prove H m () strictly ecreases with as ollows H m () = @H m @ + @H m @x x = q + q q x + c 2 + = q ( ) + q < q = q = q q xx qx 2 q xx q x q x + (x + c)q x + q x x x because q x = xq x + q x x because = e 2 (0; ) an > () xq x qx + q x + q x q xx x q x q x jq xx j 0 by (4) an (5). (EC.9) Furthermore, given m is unique, our assumption q xx < 0 implies that x m is uniquely etermine by the rst-orer conition q x ( m ; x)( m ) =. From (EC.3) an (EC.6), the monopoly pro t at the equilibrium is m ( m ; x m ) = e m ( e m ) [q( m ; x m )( m ) x m c] = e m q ( m ; x m ) 2 ( m ) ( e m ) > 0: Proo o Proposition 2: Let m (; x) enote the monopolist s iscounte pro t, the objective unction in (2). The rst-orer conitions @ m =@ = 0 an @ m =@x = 0 imply, respectively, that an optimal solution ( ; x ) to (2) must satisy: q + q q + x + c = 0 an q x = : (EC.20) Substituting (EC.20) into the e nition (EC.8) o H m (), we have H m ( ) = q q + x + c = q q (q + q q) = q < 0: (EC.2) Then, because H m ( m ) = 0 an H m () strictly ecreases in, we conclue that m <. Then x m < x ollows rom the rst-orer conitions q x ( ; x )( ) = an q x ( m ; x m )( m ) =, > 0; q xx < 0; an q x 0:
ec6 Let S m (; x) enote the sum o pro t an consumer surplus, the objective unction in (3). The rst-orer conitions @S m @ = e ( e ) 2 (q x c) + e q ( e ) = 0 an @S m =@x = 0 imply, respectively, that an optimal solution ( ; x ) to (3) must satisy q q + (x + c) = 0 an q x = : (EC.22) For any (; x) that satis es (EC.22), must be a solution to G m () q (; x s ())() q(; x s ()) + (x s () + c) = 0; where x s () is the unique solution to q x (; x) =. We will show that the unction G m () strictly ecreases with, which implies that is the unique solution to G m () = 0: By substituting q x = an then, using the implicit unction theorem, x s = = q x =q xx, we n that G m () = q + q q + (q x q x + ) xs x < q s + q x q q xx qx 2 = < 0: q xx As G m ( ) = 0, to prove that < m it remains to show that G m ( m ) < 0. Let x s enote x s ( m ). Observe that x s > x m because q x ( m ; x m ) = =( m ) > = q x ( m ; x s ) an q xx < 0: That is, holing the equilibrium evelopment time m xe, the equilibrium expeniture x m is strictly lower than the level that woul maximize the sum o pro t an consumer surplus. It ollows that G( m ) = G( m ) H( m ) < [q ( m ; x s ) q ( m ; x m )]( m ) [q( m ; x s ) q( m ; x m )] + x s x m Z x s = ( m ) q x ( m ; x)x x m = ( m ) Z x s x m q x ( m ; x)x + Z x s ( m ) x m q x ( m ; x)x + x s q x ( m ; x s ) x m q x ( m ; x m ) Z x s x m xq xx ( m ; x)x
ec7 Z x s < [ m q x ( m ; x) + xq xx ( m ; x)]x x m 0 by assumption (5): This completes the proo that < m : Proo o Proposition 3: Using = in (EC.9), at (( + ) ; x ); q q = (q x c): (EC.23) For a uopolist to be pro table, the right han sie o (EC.23) must be positive. With a Cobb- Douglas quality unction, q = q=(( + ) ), so q q = q ( + ) The uopolists are pro table i an only i " # (e ) = q > 0: (EC.24) ( + ) > z 0 = where z 0 is the unique positive solution to (e z ) = ( + )z: (EC.25) Because strictly increases in c (see the proo Proposition 6 or the general case), an > z 0 implies a lower boun on c l where c l is obtaine (EC.7) with = z=; i. e., c l = [( e z =( ) )] [( + )z] (+)=( ) ( e z ) e z ; (EC.26) ( + )z an c l is strictly positive because ( e z ) ( e z ) = ( e z ) ( + )z ( )( e z ) ( + )z > 0: (EC.27) For given ; z 0 = i an only i c c l, so c l is the threshol or non-negative uopoly pro t. Note that z 0 is invariant with respect to, an thereore (EC.26) implies that c l ecreases in. Proo o Proposition 4: For simplicity, we omit the superscript an let { i ; x i } i=;2 enote the uopoly equilibrium. The rst-orer optimality conitions or (5) an our assumption that the rms earn positive pro t at the uopoly equilibrium imply that: q( i + i ; x i ) x i q ( i + i ; x i )( i ) = c; (EC.28)
ec8 q x ( i + i ; x i )( i ) = ; (EC.29) q( i + i ; x i )( i ) x i c > 0: (EC.30) We take two steps to prove that the uopoly equilibrium is symmetric: = 2 an x = x 2. Step : We prove that i > 2 then x x 2. De ne J(xj) q (; x) [q(; x) (x + c)]q x (; x): (EC.3) By (EC.28) an (EC.29), J(x j 2 + ) = 0 an J(x 2 j + 2 ) = 0: Because 2 [0; ]; q > 0; q < 0 an q x 0; the conition > 2 implies that J(x 2 j 2 + ) J(x 2 j + 2 ) = 0: (EC.32) We next prove that J(xj 2 + ) x > 0 or all x 2 [x ; ) such that J(xj 2 + ) = 0; (EC.33) so x x 2 must be true. Otherwise we woul have a contraiction, as x < x 2 an (EC.32) woul imply existence o some x 2 [x ; x 2 ] such that J(xj 2 + ) = 0 an J(xj 2 + )=x 0. To show (EC.33), note that by (EC.29) an (EC.30) (x + c) < q( 2 + ; x ) q x ( 2 + ; x ) : (EC.34) Because q xx < 0, q(:; x) (x + c)q x (:; x) increases in x, so (EC.34) applies to all x 2 [x ; ): Substituting the let han sie o (EC.34) or (x+c) in (EC.3), or x 2 [x ; ), i J(xj 2 + ) = 0; then J(xj 2 + ) x q x ( 2 + ; x) < q ( 2 + ; x) q( 2 + ; x) : = q x q xx [q (x + c)] q x (q x ) > q x q xx [q (x + c)] = q x + jq xxjq q x q q q x by (EC.35) q q q x because J(xj 2 + ) = 0 (EC.35)
ec9 = q ( q x q + jq xxj q x q x q ) > 0 by () an (6). Step 2: We now show that i > 2 an x x 2, then (EC.28) cannot hol or both i = ; 2: Since [q( 2 + ; x ) x ] [q( + 2 ; x 2 ) x 2 ] [q( 2 + ; x ) q( 2 + ; x 2 )] (x x 2 ) [q( 2 + ; x ) q( 2 + ; x 2 )] q( 2 + ; x ) q( 2 + ; x 2 ) q x ( 2 + ; x ) because q xx < 0 = [q( 2 + ; x ) q( 2 + ; x 2 )][ ( 2 )] 0: (EC.36) Let () be the implicit unction e ne by q x (; x)( 2 ) =, holing 2 constant. It ollows that () = q x(; ()) q xx (; ()) an x = ( 2 + ). Together, q x ( + 2 ; ( + 2 ))( 2 ) = q x ( + 2 ; x 2 )( ) = ; ( 2 ) < ( ) an q xx < 0 imply that x 2 > ( + 2 ): It ollows, using q x 0, that q ( + 2 ; ( + 2 )) q ( + 2 ; x 2 ); an thereore q ( 2 + ; x )( 2 ) q ( + 2 ; x 2 )( ) [q ( 2 + ; x )( 2 ) q ( + 2 ; ( + 2 ))( ) < ( )[q ( 2 + ; ( 2 + )) q ( + 2 ; ( + 2 ))] Z 2 + = ( ) q (; ()) + q x (; ()) () = ( ) + Z 2 2 + q (; ())q xx(; ()) qx(; 2 ()) + 2 q xx (; ())
ec0 0 because q q xx q 2 x an q xx < 0: (EC.37) Combining (EC.36) an (EC.37), q( 2 + ; x ) x q ( 2 + ; x )( 2 ) > q( + 2 ; x 2 ) x 2 q ( + 2 ; x 2 )( ); so (EC.28) cannot hol or both i = ; 2. This contraiction completes Step 2 an establishes that the equilibrium must be symmetric. We now show that the symmetric equilibrium = 2, an x x = x 2 is the unique equilibrium. Motivate by the rst-orer conitions (EC:28) an (EC:29), e ne H () q (( + ); x())() q(( + ); x()) + x() + c; (EC.38) where x() is the solution to q x (( + ); x())() = ;so H ( ) = 0 an H () = ( + )q + ( + )q + (q x q x + ) x < ( + )q + (q x q x + ) ( + )q x + q x q xx = ( + )q + (q x q x + q x ) ( + )q x + q x q xx q q xx qx 2 = ( + ) + q xq x + q2 x because = q xx q xx q xx < 0; i.e., H () is strictly ecreasing in. Thereore, is unique. Furthermore, given the unique, our assumption q xx < 0 implies that x is uniquely etermine by the rst-orer conition q x ( ; x)( ) =. Proo o Proposition 5: Let (; x) enote the uopolists pro t, which is the objective unction in (7). At the optimum ( ; x ) in (7), @ @ = e ( e ) 2 [(x + c) q + ( )(( + )q + q )] = 0: (EC.39) Recall the e nition o H () rom (EC.38). Evaluating H () at an observing that x( ) = x (or brevity, we leave out arguments (( + ) ; x ) in q an q, an in an ), H ( ) = q q + (x + c)
ec = q q + q ( )[( + )q + q ] = [( + ) ]q (2 )q < 0, where the secon an thir equality employ (EC.39) an =, respectively. To unerstan the nal inequality, note that by (EC.39), [ ( ) ]q = ( )( + )q + (x + c): The let han sie is ( ) 2 q an (x + c) > 0; so the above equality implies that ( )q > ( + )q : Thereore (2 )q > ( )q > ( + )q [( + ) ]q : Since H () is strictly ecreasing an H ( ) = 0; it must be that <. Then x < x ollows rom the rst-orer conitions q x (( + ) ; x )( ) = an q x (( + ) ; x )( ) =, > 0; q xx < 0; an q x 0: The sum o pro t an consumer surplus (8): S (; x) = e q(( + ); x) e x c : An optimal solution ( ; x ) to (8) must satisy the rst-orer conitions: e @S @ = ( e ) [ q(( + ); x) + (x + c) + ( + )q (( + ); x)()] = 0 2 @S @x = e qx (( + ); x) = 0: (EC.40) ( e ) Given, let x s () enote the unique solution to (EC.40), so that x s = ( + )q x q xx. From the rst equation o (EC.40), G () q(( + ); x s ()) + (x s () + c) + ( + )q (( + ); x s ())()
ec2 satis es G ( ) = 0. Because q q xx q 2 x an q xx < 0, G () = ( + )q + ( + )q + ( + ) 2 q + [ q x + + ( + )q x ] xs = ( + )( )q + ( + ) 2 x s q + ( + )q x < ( + ) 2 q 0: qx 2 because x s = = ( + )q x =q xx q xx Thereore, in orer to prove that <, it will be su cient to show that G ( ) > 0: For brevity, let q s enote q(( + ) ; x s ( )); let q s x enote q x (( + ) ; x s ( )); let q s enote q (( + ) ; x s ( )), an let x s enote x s ( ): Then using the secon equation o (EC.40) q s x = an with a Cobb-Douglas ormulation (EC.) G ( ) = q s + (x s + c) + ( + )q s = q s + qs + c + ( + ) qx s ( + ) qs = q s + + c For brevity, let q enote q(( + ) ; x ), where x is uopoly equilibrium. Because given the same e (x ) q x = [( + ) ] = an q s = [( + ) ] (x s ) =, x x = e q s an q = s x = ( x s e ) : Let z = an e ne (; z) = q G ( z s ) ( e z ) ( + )z : Using (EC.7) or c (; z) = + + q q s ( + ) ( e z ) ( + )z
= + + ( e ) e = + e z z + ( e z ) ( + ) : e z z ( ) ( + )z ec3 Observe that i 8 2 [0; ); (; z) > 0 or all z > 0, then G ( ) > 0. To show (; z) > 0; rst note (0; z) = 0 or all z an @ @z j =0 = e z + e z z 2 z + ( e z ) = 0 @ @ j =0 = ( e z ) = e z e z so j =0 = @ @ j =0 + @ @z @z @ j =0 > 0: ( ) ( 2 e z ) ln( e z ) ln( e z ) > 0; z e z ( e z ) + + e z 2 z 2 e z z e z z It ollows that or 2 (0; ") an " su ciently small, (; z) > 0, so G ( ) > H ( ), an thereore we conclue that <. Because +, 2 ( "; ) implies that 2 (0; "), so the above also hols i is su ciently large. Finally, x < x ollows rom the rst-orer conitions q x (( + ) ; x )( ) = an q x (( + ) ; x )= =, ( ) < =, q xx < 0 an q x 0: Proo o Proposition 6: Imposing a ee-upon-sale is equivalent to increasing c or both the monopolist an the uopolists. Imposing a ee-upon-isposal is equivalent to increasing c or the monopolist because consumers ispose o the monopolist s last-generation prouct when they buy the monopolist s new prouct. Thus, the monopolist incurs the ee simultaneously with introucing a new prouct, an can eer the ee by extening the evelopment time. However, imposing a ee-upon-isposal is not equivalent to increasing c in the uopoly moel, because a uopolist incurs the ee when its competitor introuces a new prouct. In the unique uopoly equilibrium, each rm i sets ( i ; x i ) to maximize (5), in the belie that the competitor is committe to introuce a new prouct in i = units o time. Firm i knows that its choice o ( i ; x i ) cannot a ect the time until the competitor s next new prouct introuction, an hence cannot a ect the time at which
ec4 it incurs the ee. Thereore ( ; x ) remains, unchange, the unique uopoly equilibrium uner a ee-upon-isposal. To complete the proo, we must show that m, x m ; q( m ; x m ); ; x, an q(( + ) ; x ) all strictly increase with c. Reerring to H m () as e ne in (EC.8), m is the unique solution to H m () = 0 an @H m =@ < 0, so @ m @c = @H m=@ @H m =@c = @H m=@ ( m ) > 0: Given m, x m is the unique solution to the rst-orer conition q x ( m ; x m )( m ) =, an thereore @x m @c = xm m @ m @c = q x + q x q xx @ m @c > 0: The equilibrium incremental quality per new prouct q( m ; x m ) also strictly increases with c because it is a strictly increasing unction o both m an x m : By applying the same arguments with H () as e ne in (EC.38) substituting or H m (), one may conclue that ; x, an q(( + ) ; x ) all strictly increase with c: Proo o Proposition 7: As explaine in etail in the proo o Proposition 6, imposing a eeupon-sale or a ee-upon-isposal is equivalent to increasing c or the monopolist. Hence we must show that the monopolist s equilibrium pro t m ( m ; x m ) = e m [q( m ; x m )( m ) x m c] e m strictly increases with c i an only i (9) is satis e. Because @ m @x = e m [q m e m x ( m ; x m )( m ) ] = 0, (EC.4) where the secon equality comes rom (EC.5) in Lemma, m c = @m @ m @ m @c + @m @x m @x m @c = @m @ m e m @ m @c e m = e m e m e m ( e m ) 2 (q xm c) + e m @ m (q e m + q ) @c e m e m
= e m ( q + (xm + c) e m = e m @ q m e m ; @c + q + q ) @ m @c ec5 where the nal equality ollows rom (EC.4) in Lemma. It ollows that the monopolist s pro t strictly increases with c i an only i q @ m =@c > 0: (EC.42) (We establishe in Proposition 6 that @ m =@c > 0.) Using the implicit unction theorem, i erentiating c = q m in (EC.2), @ m =@c = q + q + q x x + q m m m Applying (EC.3) or x m = m ; the above equals (q + q ) = q = q + q m + q ( ) m + m + q m + m m 2. Substituting the above expression or @ m =@c strictly increases with c i an only i into (EC.42), we n that the monopolist s pro t < m 2 : (EC.43) The above is true when m = 0, alse when m =, an the erivative o the right-han sie m 2 2 2 m
ec6 = 2 = 2 = < 0 + m 2 + m 2 + ( ) 2 ( + ) 2 m so there exists a unique m proit > 0 at which the two sies o (EC.43) are equal an (EC.43) hols at m i an only i m < m proit. As @ m =@c > 0, there exists some c m proit > 0 such that m < m proit; an thus m =c > 0 i an only i c < c m proit. Analogous arguments apply or the uopoly moel. As explaine in etail in the proo o Proposition 6, imposing a ee-upon-sale is equivalent to increasing c or uopolists. Thereore we must show that the uopolists pro t = e e [q(( + ) ; x )( ) x c] increases with c i an only i (20) is satis e. The uopolist s rst-orer conition or expeniture (EC.0) implies that Thereore, @ @x = e e [q x(( + ) ; x )( ) ] = 0: c = @ @ @ @c + @ @x @x @c = @ @ e @ @c e = " e e e ( e ) 2 (q x c) + e = e ( e ) = e ( e ) e (( + )q + q ) q + q e + ( + )q + q @ @c ( [ ( + )e ]q + (2 )q @ e @c # ) @ @c e e where the next-to-last equality ollows rom the uopolist s rst-orer conition or (EC.9) an = = e : It ollows that the uopolists pro t strictly increases with c i an only i [ ( + )e ]q + (2 )q > ( e ) : (EC.44) @ =@c
ec7 (We know rom Proposition 6 that @ =@c > 0.). From (EC.8) @q @ + q x x = q + From (EC.7) in Lemma 2, (@q=@ ) + q q xx @q @ =@c = @ + q x x ( + ) + ( + )( ) 2 ( + ) q = q + ( + ) + ( + )( ) 2 ( + ) q = ( ) q + ( + )( ) 2 + ( )( + ) ( + ) 2 = ( ) q + q ( + )( ) 2 + + ( + ) + q = ( ) q + ( ) q q q By substituting the right han sie above or, (EC.44) is equivalent to @ =@c [ ( + )e ] e > 2 ( ) ( + ) + (2 ) ( e ) 2 + + ( + )( ) 2 ( e ) ( + ) + + : By enoting z =, iviing both sies by, an rearranging terms, the above inequality simpli es into (2 e z ) ( e z ) 2 ( + )z 2 e z ( e z ) e z + > 0: (EC.45) ( + )z Let g(z) enote the let han sie o (EC.45). The uopolists are pro table at the equilibrium an thereore, rom (EC.25), that or rms to be pro table at the equilibrium, z 0 where ( + )z 0 e z 0 = :
ec8 It is easy to veriy that g() < 0 an g(z 0 ) = 2 = 2 = > 0: e z 0 ( e z 0 ) 2 ( + )z0 2 e z 0 e z 0 z 0 e z 0 z 0 ( + )z 0 ( )(e z 0 ) e z ( e z 0) 0 e z 0 + ( + )z 0 + e z 0 We will prove that there exists a unique point z proit 2 (z 0 ; ) such that g(z proit) = 0 an g(z) > (< )0 or z < (>) z proit. (Prior arguments show that this will imply that the uopolists pro t strictly increases with c i an only i < z proit=.) It will be su cient to veriy that g 00 (z) < 0 which is true because in (EC.45), components are all convex unctions o z: e z ; ( e z ) 2 e z ; ez z 2 z, an e z ; z We conclue that the uopolists pro t strictly increases with c i an only i < z proit=: (EC.46) Because strictly increases with c by Proposition 6; (EC.46) is equivalent to c < c proit = C(z proit) where C (z) is e ne by (EC.7) using = z=. Observe that C (z) strictly ecreases with. As z proit is invariant with respect to, this establishes that c proit strictly ecreases with. Now suppose + = ; in which case (EC.43) can be simpli e to m < z m proit = ln[+=( )], an the let-han sie o (EC.45), g(z), becomes g(z) = 2 e z ( e z ) ( + )z [( )ez + ]
ec9 Because g(z m proit) = g(ln[ + =( )]) [ + =(2 )]=( + ) = 2 2 ln[ + =( )] (3 2) ln[ + =( )] ( )=( + ) = (2 ) ln[ + =( )] (3 2) ln[ + =( )] 2 + (2 ) ln[ + =( )] > 0; an g(z) 0 or z z proit, z m proit < z proit. As we have shown previously, increases in c, so C (z) increases in z an thus C (z proit) > C (z m proit): To show c m proit < c proit, it is su cient to prove that c m proit < C (z m proit). For brevity enote z m proit by z, rom (EC.7), when = ; C (z) = [( e z )] =( ) [( + )z] ( e z ) ( ) e z (2 )=(+) z [( e z =( ) )] z z ( ) )=(+) e z (2 > [( e z =( ) )] [e z + z z ( )] (2 )=(+) [( e z =( ) )] ( )[z ( e z )] (2 )=(+) = c m proit by using (EC.2) with = an m = m proit = z=. Proo o Proposition 8: Social welare at the uopoly equilibrium is W = e q(( + ) ; x ) e x c e (k + z) ; note that a ee-upon-sale is a transer rom the uopolists to other sectors o the economy (collection an recycling), so it oes not irectly a ect social welare. Proposition 6 establishe that imposing a ee-upon-sale strictly increases : At the uopoly equilibrium, W = e q ( e ) 2 + e ( + )q e e = ( e ) 2 x c (2e e 2 )(k + z) + ( q x ) x q + (x + c) + e (2 e x )(k + z) + ( + )q +
ec20 e = e (2 e x )(k + z) + q ( e ) 2 + > 0: because o (EC:28) (EC.47) Hence imposing a ee-upon-sale strictly increases social welare in the uopoly moel. Social welare at the monopoly equilibrium is W m = e m q( m ; x m ) e m x m c e m (k + z) ; (EC.48) note that a ee-upon-sale or a ee-upon-isposal is a transer rom the monopolist to other sectors o the economy (collection an recycling), an oes not irectly a ect social welare. In the monopoly moel, imposing a ee-upon-sale or ee-upon-isposal is equivalent to increasing c. Proposition 6 tells us that imposing either a ee-upon-sale or a ee-upon-isposal will strictly increase m. Thereore imposing a small ee-upon-sale or ee-upon-isposal strictly increases social welare i the ollowing quantity is positive @W m @ m = = @ m @c + @W m @x m @x m @c e m ( e m ) [( q + 2 (xm + c) + q ) + e m (2 e m )(k + z) e m ( e m ) 2 qx ( e m )] xm @ m m @c + q q + xm m + e (2 e m )(k + z) m @ m @c ; where the nal equality ollows rom (EC.4), (EC.5), an + =. Assuming a Cobb-Douglas quality unction, the expression above is strictly positive i an only i q q < xm + e (2 e )(k + z) = m (q + q ) + e m (2 e m )(k + z); (EC.49) where the equality ollows rom (EC.3) in Lemma. Note that m is invariant with respect to (k + z), an the right-han sie o (EC.49) strictly increases in (k + z) an converges to as (k + z)!. Hence there exists a nite constant k, such that a ee-upon-sale or ee-upon-isposal strictly increases social welare i (k + z) > k but strictly ecreases social welare i (k + z) < k.
ec2 Furthermore, the constant k is strictly positive i the inequality in (EC.49) is violate at k + z = 0, that is, i q q (q + q ); which is true i an only i (using q = q= m in the above an with a simple transormation), m $m welare (EC.50) where $ m welare is implicitly e ne by ( e $ ) $ + ( + e $ ) = : (EC.5) Because m strictly increases with c by Proposition 6, (EC.50) is equivalent to! c c m ) =( m $ welare = ( e welare) =( ) e $m welare =( )+ where c m welare is e ne by (EC.2) with $ m welare = m welare: $ m welare $ m welare; We now prove the corollary, state in the paper immeiately ater Proposition 8, that c < c m welare implies that x m > c. From (EC.49), q q( ): From (EC.2) c = q = ( )q q m 2. By canceling out q in the above, c ( )q q( ) = q = x m q x = x m x m : Finally, note that our e nition (9)-(23) o social welare oes not inclue the environmental, health an en-o-lie processing costs or the generation zero prouct that is introuce at time zero an ispose at time. (This is consistent with the act that we o not inclue the R&D an prouction cost or the generation zero prouct in calculating manuacturers pro ts an social welare, an the investment in esign or recyclability to reuce k, incorporate in Section 6, oes
ec22 not a ect the generation zero prouct.) A ee-upon-sale postpones the isposal o the generation zero prouct an so reuces its iscounte environmental, health an processing cost, as oes a ee-upon-isposal in the monopoly moel. Proo o Proposition 9: In the uopoly moel, consumer surplus is CS = e q(( + ) ; x ) e q(( + ) ; x )( ) = e 2 ( e ) q(( + ) ; x ): (EC.52) Proposition 6 shows that imposing a ee-upon-isposal has no e ect on the uopoly equilibrium ( ; x ), an thereore the ee-upon-isposal oes not a ect consumer surplus. Proposition 6 also shows that imposing a ee-upon-sale strictly increases, so or a small ee-upon-sale to increase consumer surplus, it is necessary that " CS = e 2 @q ( e ) @ + q x x q!# 2 e 0 e which is true i an only i (note q x = an = ( e )=), (2 e )q @q @ x = @q @ + q where the last equality comes rom (EC.8). By substituting @q=@ = q= an canceling q rom both sies, 2 ( ) : Since < an, the above inequality requires >, which cannot be true because the rst-orer conition (EC.9) implies that q q = (q x c): To enable the uopolists to earn a pro t, q x c > 0 an thereore < q q = ( + ) <.
ec23 In the monopoly moel, consumer surplus is CS m = e m q( m ; x m ) e m q( m ; x m )( m ) = e 2 m ( e m ) q( m ; x m ): (EC.53) A slight increase o equilibrium evelopment time m changes consumer surplus by CS m = e 2 m x q m m ( e m + q x ) m 2 q e m e m : (EC.54) Proposition 6 shows imposing a ee-upon-sale or ee-upon-isposal strictly increases m ; so consumer surplus strictly ecreases i q + q x x m m 2 q e m e m < 0: Using (EC.3) in Lemma or x m = m, the let-han sie o the above is q + (q + q ) ( ) e m q 2 = q e m + q = q + e m ( )( e m ) q: e m 2( ) ( )( e m ) 2 e m e m Because q = q= m, the expression above is strictly negative i an only i ( e m ) + e m < 2( ): (EC.55) m Because its let-han sie strictly ecreases in m an m strictly increases in c, there exists some c m cs such that (EC.55) is satisie i an only i c > c m cs. To emonstrate (EC.55) is always true when c > x m, apply Cobb-Douglas ormulation an substitute q with q= m an q x with q=x m in the rst-orer conitions (EC.4) an (EC.5), m = x m + c q < 2x m q = 2x m q q x = 2: Insert the above into (EC.55) ( e m ) + e m < 2 + e m < 2( ): m
ec24 Proo o Proposition 0: Uner collective or iniviual EPR, the monopolist, as the only rm in the market, bears the en-o-lie cost or its own proucts. Thereore the monopolist chooses the en-o-lie cost k to max k2( c;k] where L( m ; x m ) I(k) + L( m ; x m e m ) k ; e m e m [q( m ; x m )( m ) x m c] e m an ( m ; x m ) is the unique monopoly equilibrium evelopment time an expeniture corresponing to en-o-lie cost k, characterize by the rst-orer conitions (EC.6) an (EC.7) with (k + c) substitute or c. Our assumptions that I(k) is strictly convex or k 2 ( c; k] an satis es lim k# c I 0 (k) = an I 0 (k) = 0 guarantee existence o a unique interior solution k m 2 ( c; k) at which I 0 (k m ) = e m L e m k = e m L m e m m k : (EC.56) Since L= m > 0 by Proposition 2 an m =k > 0 by Proposition 6, L=k > 0. Holing m constant at the level corresponing to the monopolist s optimal en-o-lie cost k m, e ne k = arg min I(k) + e m k : k2( c;k] e m Our assumptions that I(k) is strictly convex or k 2 ( c; k] an satis es lim k# c I 0 (k) = an I 0 (k) = 0 imply that I 0 (k ) = e m ; (EC.57) e m an (EC.56) implies that I 0 (k m ) < As I() is strictly convex, (EC.57) an (EC.58) imply that e m : (EC.58) e m k m > k :
ec25 In both the monopoly an uopoly moels, the ee-upon-sale ARF is invariant with respect to the en-o-lie cost k an thereore the rms have no incentive whatsoever to invest in reucing the en-o-lie cost below its maximum k = k. Iniviual EPR is a ee-upon-isposal, as is collective EPR with total cost allocate accoring to share o proucts ispose by consumers. Hence by Proposition 6, the uopoly equilibrium evelopment time an expeniture are invariant with respect to the en-o-lie cost. Uner collective EPR with total cost allocate accoring to share o proucts ispose by consumers, each uopolist bears the en-o-lie cost or its own prouct at the time that the competitor introuces a new prouct. Our assumption that consumers are homogeneous implies that all consumers ispose o the last-generation prouct upon purchasing the new prouct, an thereore no mixing o proucts occurs in the return stream. This makes collective EPR with total cost allocate accoring to share o proucts ispose by consumers precisely equivalent to iniviual EPR. Thereore each uopolist maximizes its pro t by choosing the en-o-lie cost to minimize the sum o investment an iscounte en-o-lie cost. Uner collective EPR with current-sales-base allocation, the en-o-lie cost or a uopolist s prouct is pai by the competitor when the competitor introuces a new prouct an causes consumers to ispose o the uopolist s prouct. Thereore any investment by a uopolist to reuce the en-o-lie cost o its prouct will only bene t the competitor by lowering its new prouct introuction cost. Moreover, the uopolist s pro t eclines with the en-o-lie cost or its prouct because this e ectively reuces its competitor s cost to introuce a new prouct an, by (5), causes the competitor to introuce new proucts more quickly. Thereore the uopolist has a is-incentive to invest in reucing the en-o-lie cost below its maximum k = k.
ec26 Extening the Strategy Space to Allow Each Duopolist to Introuce Multiple Consecutive New Proucts: In the uopoly moel ormulation in 3, we restricte the strategy space so that each rm coul introuce at most one new prouct in the expecte time winow between consecutive new prouct introuctions by its competitor. That is, we assume that the rms woul alternate in new prouct introuction. We now assume = 0 an allow the uopolists to introuce new proucts at any time. In particular, a rm may introuce arbitrarily many new proucts in the expecte time winow between consecutive new prouct introuctions by the competitor. The incremental quality or a new prouct is q(; x) where is the time elapse since the last new prouct introuction by either rm an, as in Section 3, x is the evelopment expeniture or that new prouct. Proposition EC. establishes that the unique stationary alternating equilibrium erive in Section 3 is a sequential equilibrium in the generalize state space. In the sequential equilibrium, each rm chooses to introuce exactly one new prouct between the consecutive new prouct introuctions by its competitor, i.e., chooses to alternate with the competitor rather than introuce multiple new proucts in a row. Proposition EC. Suppose that = 0 an the uopolists may introuce new proucts at any time. The solution ( ; x ) to (5) characterizes a sequential equilibrium in which the uopolists alternate in new prouct introuction, each uopolist chooses evelopment time 2 an expeniture x or each new prouct, an the time between consecutive new prouct introuctions is. Proo o Proposition EC.: We will show that i a rm were to eviate rom the alternating equilibrium ( ; x ), then that rm woul reuce its iscounte pro t an it woul be sequentially rational or both rms to revert to the alternating equilibrium ( ; x ) in the continuation game. Without loss o generality, we can assume that rm introuces a new prouct at time 0, assume that rm 2 is committe to introuce a new prouct at time, an consier eviation by rm rom its equilibrium strategy uring the time perio [0; 3 ]. (Recall that the equilibrium strategy has rm introuce a new prouct at time 2 an rm 2 introuce a new prouct at time 3.) Consumers expect rm 2 to introuce a new prouct at time ; so Lemma 3 implies that rm
ec27 will earn strictly negative pro t on any new proucts introuce uring (0; ]. By introucing new proucts uring (0; ], rm cannot in uence its own pro t or the other rm s pro t in the continuation game rom time. Thereore introucing one or more proucts uring (0; ] can only strictly reuce rm s iscounte pro t. Lemma 3 There exists 0 2 ( ; 2 ) such that < 0 or < 0 max q(; x)( ) x cg 2[0;] x0 0 or 0 (EC.59) At time rm 2 introuces its new prouct an, ollowing its equilibrium strategy, commits to introuce its next new prouct at time 3. Consumers expect rm 2 to introuce a new prouct at time 3. Thereore Lemma 3 implies that i rm introuces a new prouct at any time t 2 (3 0 ; 3 ), rm will earn strictly negative pro t on any aitional new proucts that it introuces uring (t; 3 ]: Proposition 4 establishe that i rm introuces a single prouct uring [ ; 3 ], it must o so at time 2 or earn lower iscounte pro t. It remains to prove that rm cannot introuce a prouct at time t 2 ( ; 3 0 ], introuce secon prouct uring (t; 3 ] an earn positive pro t on both those two proucts. In proving this result, we can assume that rm introuces no prouct uring (,t). A prouct introuce uring (,t) woul earn strictly negative pro t by Lemma 3 (or that prouct, < t 2 0 < 0 ) an woul also strictly reuce the incremental quality an hence pro t or the prouct introuce at time t. Let 0 an 00 enote the optimal intervals or rm s two new proucts introuce uring ( ; 3 ]; so rm introuces the rst prouct at time + 0 an the secon one at time + 0 + 00 where 0 < 0 2 0 an 0 < 00 2 0. With the corresponing optimal expenitures x 0 an x 00, the incremental quality o the two proucts are q( 0 ; x 0 ) an q( 00 ; x 00 ) respectively. A sequential equilibrium requires consumers belies to be consistent with the optimal actions o rm ater it eviates rom the equilibrium. By Lemma 3, + 0 3 0 < 2, so the consumers see rm eviate rom its equilibrium strategy when it introuces the new prouct at time + 0 an, rom (EC.59), the consumers realize that rm will optimally introuce another new prouct at time + 0 + 00 3. Thereore consumers are willing to pay q( 0 ; x 0 )( 00 ) or the rst prouct
ec28 an q( 00 ; x 00 )( 000 ) or the secon prouct where 000 = 2 0 00. Because x 0 an x 00 maximize q( 0 ; x 0 )( 00 ) x 0 an q( 00 ; x 00 )( 000 ) x 00 respectively, the corresponing rst-orer conitions an properties o the Cobb-Douglas quality unction (EC.) imply that q x ( 0 ; x 0 )( 00 ) = ( 0 x 0 ) ( 00 ) = an q x ( 00 ; x 00 )( 000 ) = ( 00 x 00 ) ( 000 ) = : It ollows that x 0 = [( 00 )] 0 ; q( 0 ; x 0 ) = [( 00 )] 0 x 00 = [( 000 )] 00 ; q( 00 ; x 00 ) = [( 000 )] 00 so that q( 0 ; x 0 )( 00 ) x 0 = ( )q( 0 ; x 0 )( 00 ) = ( ) [( 00 )] 0 q( 00 ; x 00 )( 000 ) x 00 = ( )q( 00 ; x 00 )( 000 ) = ( ) [( 000 )] 00 : (EC.60) From uopoly rst-orer conition (EC.7), with = an = 0, c > ( ) [( )] ( ); where ( ) = =( e ) =( ), so or both new prouct introuctions to be pro table, q( 0 ; x 0 )( 00 ) x 0 > c > ( ) [( )] ( ) q( 00 ; x 00 )( 000 ) x 00 > c > ( ) [( )] ( ): By (EC.60), this means min[( 00 )] 0 ; [( 000 )] 00 g > [( )] ( ): De ne y 0 = 0 ; y 00 = 00 ; y 000 = 000 ; an y =, the above inequality is equivalent to min( e y00 ) y 0 ; ( e y000 ) y 00 g > ( e y ) y (y ):
ec29 To prove that the two new proucts cannot both be pro table, it su ces to prove that the above cannot hol by showing or any y 0 ; y 00 ; y 000 such that y 0 + y 00 + y 000 = 2y, max (y 0 ;y 00 ;y 000 ):y 0 +y 00 +y 000 =2y min( e 00) y y 0 ; ( e y000 ) y 00 gg < ( e y ) y (y ): (EC.6) The maximum on the let-han sie is always achieve with equality between the two terms within the min operator. ( e y00 ) y 0 = ( e y000 ) y 00 ; (EC.62) to unerstan this, observe that i the two terms were not equal, one coul increase the minimum o the two by increasing either y 0 or y 000 an simultaneously reucing the other until both terms were equal. It ollows that (EC.6) is true i an only i or all y 0, y 00, an y 000 that satisy (EC.62), ( e y00 ) y 0 + ( e y000 ) y 00 < 2( e y ) y (y ) (EC.63) The rst step to prove (EC.63) is to show that or all y 0, y 00 an y 000 satisying (EC.62), ( e y00 ) y 0 + ( e y000 ) y 00 2( e y ) y (y ) ( e 00)y y 0 + ( e y000 )y 00 ; (EC.64) 2( e y )y (y ) which is immeiate i y 00 y an y 000 y so that!! e y00 e y00 e y e an e y 000 y e ) y e y 000 e : y I this is not the case, then because y 0 > 0 an y 0 + y 00 + y 000 = 2y, at most one o y 00 an y 000 can be greater than y. Without loss o generality, let y 000 < y < y 00. Due to (EC.62), or all 0 e, ( e y00 ) =( e) y 0 y = ( e y00 =( e) ( e 000) ) y 00 ( e y00 ) = ( e y000 ) =( e) y 00 ( e y ) e y00 000 e ( )( e) ( e y000 ) =( e) y 00 : (EC.65) Fix y 0, y 00, an y 000, an let L(e) = 2! 4y 0 e y00 2y (y ) e y e + y 00 e y000 e y! e 3 5
ec30 For 0 e ; L(e) e = 2y (y )( e) 2 y 00 2 4y 0 e y00 e y e y 2y (y )( e) 2 y00 e y! e! 000! ln e y 00 + y 00 e y 000 e y e y! e "!!# e ln e y00 + ln e y 000 e y e y where the inequality ollows rom (EC.65). Because y 00 + y 000 = 2y y 0 < 2y,!! ln e y 00 + ln e y 000 e y e y max 0<yy 00 ( e y ln + ln e (2y e y e y so L(e)=e < 0 (0 e ), which establishes that (EC.64) is true. Uner (EC.64), it su ces to prove (EC.63) by showing y)! 3 ln e y 000 5 e y!) < 0; max y 0 ; y 00 ; y 000( e y 00 )y 0 + ( e y000 )y 00 g < 2( e y )y (y ): (EC.66) The rst orer conitions or the maximization problem on the let han sie o (EC.66) are e y00 = ; (EC.67) y 0 e y00 + ( e y000 ) = ; (EC.68) y 00 e y000 = (EC.69) where is the Lagrangian multiplier associate with the constraint y 0 + y 00 + y 000 = 2y. Eliminating rom (EC.67) an (EC.69), y 000 = ln[( e y00 )=y 00 ]. Substituting that expression into (EC.68) an using (EC.67) to eliminate establishes that y 0 = (e y00 )=y 00 an it ollows that ( e y00 )y 0 + ( e y000 )y 00 = ey 00 2 + e y00 y 00 + y 00 + 2e y 00 2: Because y = y 0 + y 00 + y 000, y = (y 0 + y 00 + y 000 ) = 0:5 " e y00 + y 00 ln e y y 00 y 00 00!# an thereore e y = s e y00 e y 00 +( e y 00 )=y 00 y 00 2
ec3 Recall (y ) = =( e y ) =(y ), so 2( e y )y (y ) = 2y 2 + 2e y = ey 00 + y 00 ln e y y 00 y 00 00! 2 + 2 s e y00 e y 00 +( e y 00 )=y 00 y 00 2 : Thereore, or y 00 > 0; 2( e y )y (y ) ( e y00 )y 0 + ( e y000 )y 00 s! e = 2 y00 e y 00 +( e y 00 )=y 00 y 00 2 ln e y00 y 00 2e y00 + e y 00 y 00 (EC.70) > 0: To veriy the last inequality, notice that ln[( e y00 )=y 00 ] increases in y 00, + 2e y00 ecreases in y 00 ;an ln[( e 8 )=8] > + 2e 8, so (EC.70) is strictly positive when y 00 8. To veriy that (EC.70) is strictly positive or y 00 2 (0; 8); one may simply plot its value as a unction o a single variable y 00 over [0; 8]. Thereore, (EC.66) is true, so (EC.63) an (EC.6) are true. This completes the proo that rm cannot introuce a prouct at time t 2( ; 3 0 ], introuce secon prouct uring (t; 3 ] an earn positive pro t on both those two proucts. We conclue that by eviating rom its equilibrium strategy uring [0; 3 ], rm can only reuce its iscounte pro t an cannot in uence its own pro t or rm 2 s pro t in the continuation game rom time 3. By analogous arguments, it is sequentially rational or both rms to use their equilibrium strategies in the continuation game rom time 3 even ater a eviation by rm uring [0; 3 ]. We conclue that the alternating equilibrium ( ; x ) is a sequential equilibrium. Proo o Lemma 3: De ne 0 = in : max 2[0;] x0 q(; x)( ) x cg 0g an observe that 0 < 0 < 2
ec32 because q(0; x) = (0) = 0 an the alternating equilibrium ( ; x ) is pro table or the uopolists, so q( ; x )( ) x c > 0. As both q(; x) an () are continuous an strictly increasing unctions o ; or any > 0, max q(; x)( ) x cg > max q(; x)( 0 ) x cg = 0: 2[0;] x0 2[0; 0 ] x0 This completes the proo o (EC.59). It remains to prove that 0 > by showing that max q(; x)( ) x cg < 0: (EC.7) 2[0; ] x0 Let (b; bx) be an optimal solution to (EC.7), bq = q(b; bx) an bq = q (b; bx). The rst-orer conition bq ( b) bq ( b) = 0 implies that ( b) = bq bq + bq, so = b + ln + bq : bq Substituting the expression or q or Cobb-Douglas quality unctions rom (EC.), the above equalities simpliy to = b + ln + b bx = q x = [( b)] =( ) b because q x (b; bx) = (EC.72) b = ( b); (EC.73) bx an bq = [( b)] =( ) b; (EC.74) rom (EC.6) an the properties (EC.) o the Cobb-Douglas quality unction. Since by (EC.73) an (EC.74), q(b; bx)( b) bx = ( ) =( ) [( b)] =( ) b: From (EC.7) with = an = 0, an because ( b) < ( ), it is su cient to prove q(b; bx)( b) bx < c by showing b ( )
ec33 or equivalently that (b + )( e ) : To establish the above inequality, e ne w = b=( ), so by (EC.72), = b + ln + b = ( )w + ln( + w) e ( e = + w an (b + )( e e ( )w ) = [( )w + ] + w = ( )w + e ( )w w + w )w Because or any w 0, e w [ w=( + w)]g = we w w + 0 + w so [ e ( )w ( +w w)] is maximize at = 0 when the value becomes e w. Thus (b + )( e ) = ( )w + e ( )w w + w ( )w + e w < ( )w + ln( + w) = ; which completes the proo.
ec34 Construction o the Numerical Examples in 5 In constructing the numerical examples in 5, we assume that the quality unction is o the unctional orm q(; x) = vx. Imposing a ee per unit sale increases non-r&d cost per new prouct introuction rom c to c + N where N is the number o units sol per new prouct generation. We use publicly available ata to t the parameters v, ;, c, an N; as escribe below. In both monopoly an uopoly moels, the rst-orer conition or equilibrium R&D expeniture is q x = : The Cobb-Douglas unctional orm o q(; x) implies that xq x = q, so = x q : (EC.75) The enominator q is the revenue generate by a new prouct, an thereore is the ratio o R&D expense to revenue, which we obtain rom manuacturers nancial statements. In the monopoly moel, the rst-orer conition or equilibrium evelopment time is q q (x + c)= = 0: Using the properties o q(; x) that xq x = q an q = q, the above equation is easily transorme into = () x + c q (EC.76) where the term (x + c) q (EC.77) is the operating margin or the new prouct. Thus we can compute rom public ata on operating margins an the time between new prouct introuctions (). We take an analogous approach in tting the uopoly moel. We transorm the rst-orer conition or the equilibrium q q (x + c) = 0
ec35 into = () x + c q () : (EC.78) In all examples we consiere, the Cobb-Douglas requirement that + implie that 0, an thereore we set = 0: Straightorwar manipulation o the unctional orm we assume or the quality unction gives v = q x = q x ; (EC.79) where the numerator q is the revenue per new prouct, which we obtain rom manuacturers nancial statements. Furthermore, having alreay obtaine, q, an, an recalling that x = q rom (EC.75), we can then compute the value o v:given revenue q an R&D expeniture x, we can also etermine non-r&d cost c by c = ( operating margin)*q x. (EC.80) MP3 Player Category: We use the monopoly moel or this prouct because Apple ipo has market share in excess o 70% (Hall 2006). Ater introucing the rst ipo in 200, Apple has been launching new versions o the prouct an retiring ol ones rom the market every year (Apple-0K, 200-2007), so we set m = year. Sales an revenues grew rapily as the ipo penetrate the market in 200-2005, but show signs o stabilizing in 2006 an 2007. As our moel assumes stationarity, we use nancial ata collecte since the prouct introuction in September 2006, which we believe is best represents the uture, maturing market. We are unable to n ipo-speci c ata on R&D expense an operating margin, an thereore we assume that R&D expense as a percentage o revenue an operating margin are 3% an 34%, respectively, corresponing to the overall gures or Apple (Apple 2007). Then (EC.75) an (EC.76) imply that = 0:03 an = 0:36.
ec36 ipo revenue growth (stablizing in the last year) 9000 8000 7000 6000 5000 4000 3000 2000 000 0 200 2002 2003 2004 2005 2006 2007 2008 In scal year 2007, ipo unit sales is N = 5:6 million. The total revenue is $8:305 billion, which implies R&D expeniture x = q = $249 million. From (EC.79) with = an (EC.80), v = 4934 0 6, an c = $5:232 billion. Workstation Category:. The workstation category is also best represente by our monopoly moel, because introuction o new workstations is riven by Intel s launching o new microprocessors in its Xeon line, an Intel has more than an 85% market share among proucers o such microprocessors or workstations (Dignan 2007). Between Intel an workstation en-users are the workstation assemblers, such as Sun, Dell, an HP. The assembler s market is highly competitive an pro ts are slim. Intel is responsible or a majority o the R&D investment in the workstation prouct category, an captures a majority o the associate pro t generate by new prouct introuction. Thereore, espite the complexity o the workstation supply chain, our simple monopoly moel o Intel s new prouct introuction process is a reasonable approximation. Ater introucing Pentium III Xeon in 999, Intel launche Xeon MP in 2002 base on Pentium IV technology, an the Duo-Core an Qua-Core Xeons in March 2006. Thus we set m = 3 years. We are not able to n public inormation on Intel s R&D expenitures or workstation (Xeon) microprocessors, an thereore take nancial ata or the company as a whole as representative. We use accumulate totals to average out small year-to-year variations. From 999 to 2006, Intel
ec37 generate $254:98 billion in total revenue an spent $34:99 billion on research an evelopment (Intel-0K, 999-2006), so rom (EC.75) = 34:99 254:94 = 0:37: Fortunately, we i obtain more prouct-speci c inormation about the operating margin. Intel s workstation microprocessor business is a part o the Digital Enterprise Group (DEG), or which revenue an operating income are broken out in Intel s annual reports (Intel-0K, 2003-2006). For the our nancial years 2003-2006, DEG s total revenue is $92:85 billion an total operating income is $30:4 billion, which translates into an operating margin o 32:5%: Using (EC.76) with = 3, = 0:387: As Intel oes not release speci c revenue gures or its workstation (Xeon) microprocessor business, we estimate that revenue rom scattere ata on quarterly sales o workstations an microprocessor prices. Reports rom a computer inustry consulting rm, Jon Peie Research, put the total number o workstations shippe in 2Q 2006 as 67; 000 units or 2Q 2006 (JPR 2006), 573; 000 units or 4Q 2006, an 674; 000 or Q 2007 (JPR 2007). Thereore we assume average quarterly sales o 600; 000 microprocessors, which translates, over the 3-year prouct lietime, into sales o N = 7:2 million units per new prouct. Depening on processor variety, Xeon s retail price can range rom $200 to $000 or more (e.g. see price review an comparison at websites like www.ealtime.com): We use the meian value o $500 to set total revenue per new workstation processor q = 7:2500 = $3:6 billion. It ollows that R&D expeniture is x = q = $494 million. From (EC.79) with = 3 an (EC.80) v = 59:88 0 6 an c = $:936 billion. Vieo Game Console Category: In the vieo game console market, Sony s Playstation battles Microsot s Xbox. Committe gamers are the primary target consumers or each new generation o the Sony Playstation an Microsot Xbox, whereas Ninteno has introuce a line o Wii consoles to
ec38 appeal to a i erent emographic - casual or rst-time gamers (Casey, 2006). Due to this separation in target consumer populations, we t our uopoly moel to represent the competition between Sony an Microsot. The competition starte in 200 when Microsot entere console market by introucing Xbox (Microsot, 200). Sony respone by introucing PlayStation2 Portable (PSP) in 2004 (SONY-20F, 2004). Microsot introuce Xbox360 in 2005 (Microsot, 2005), ollowe by Sony s introuction o PlayStation3 in 2006 (SONY-20F, 2006). Neither rm release a new prouct in 2007. Base on the average time between new prouct introuctions, we set the uopoly equilibrium = :5 years: We obtain R&D expense an operating margin rom a nancial report or Sony s game business le with the SEC. In Microsot s annual report, the vieo console belongs to Home an Entertainment segment, within which the nancial ata is primarily etermine by the sotware an is thereore not representative o the vieo game console business. From 2002 (when Microsot entere the market) to 2007, Sony s game segment has incurre an R&D expense o 468:2 billion yen an generate a total revenue o 5444 billion yen with operating income o 82:8 billion yen (Sony-20F, 2002-2007), so R&D expense is 8:6% o the total revenue an the operating margin is :5%. The slim operating margin is consistent with reports on Microsot that its vieo game consoles are unpro table. Thereore we use the gures rom Sony. From (EC.75) an (EC.78) with = :5, = 0:086 an = 0:95. Our moel assumes that users are homogeneous an thereore buy a new prouct at the time o its introuction. In reality, buyers o vieo game consoles are heterogeneous (an extension we aress in 7) an thereore buy graually over time as the manuacturer marks own prices. Thus we estimate total unit sales base on proucts that have alreay been retire, an o not use ata or new proucts that are in the mile o the selling cycle. Speci cally, we set total sales o a vieo game console at N = 20 million units corresponing to the total sales o Sony s PSP an Microsot s Xbox. The price o a vieo game console can range rom slightly below $250 to $600
ec39 (Kageyama, 2006), so we an approximate meian price o $400 to compute the revenue per new prouct q = $400 20 million = $8 billion an then R&D expeniture x = q = $0:48 billion. By (EC.79) with = :5 an (EC.80), v = 698 0 6 an c = $7:92 billion Mobile Phone Category: We take a uturistic view o the U.S. mobile phone inustry. Currently in the US, mobile phones are commonly bunle with a service agreement; service proviers subsiize the phone purchase an recover revenue through subsequent service charges. However, the inustry tren is to unbunle phone sales rom service so that, in the near uture, US consumers will buy mobile phones at market prices, inepenently o their choice o service provier (Segan, 2007; Holson, 2007; Searcey, 2007). We t our uopoly moel to this uture unbunle scenario or the US market an etermine parameter values by a i erent process rom the above. We assume in this uture scenario, consumers will upgrae their mobile phones as requently as they o toay. The average liespan o a mobile phone is 8 months (NY report 2004), which translates into = :5 years: We ocus on US market o GSM phones where Nokia an Motorola are ominant. In the past three years, the two rms reporte comparable R&D expense as percentage o total revenue ( 0%) an operating margin (0% 5%) or the entire company (income statements are at nance.yahoo.com). Nokia releases quarterly nancial inormation that report ata speci c or mobile phones (Nokia, 2004-2007, no such etaile inormation is given in Motorola s reports), which we use to compute an. From Q 2004 to 3Q 2007, the total revenue o Nokia s mobile business is e87.746 billion, R&D expense is e4:673 billion, an operating income is e5:60 billion. This means R&D expeniture is 5:7% o total revenue an the operating margin is 8:4%:From (EC.75) an (EC.78) with = :5, = 0:057 an = 0:946.