Technical Appendix for: Complementary Goods: Creating, Capturing and Competing for Value

Transcription

1 Technical Appendix for: Complementary Goods: Creating, Capturing and Competing for Value February, 203

2 A Simultaneous Quality Decisions In the non-integrated case without royalty fees, the analysis closely follows Proof of Proposition with the only difference being solving for the first order conditions for quality decisions simultaneously instead of sequentially; yielding equilibrium qualities α S = and 3 2 k βs =, where 3 2 k superscript S denotes simultaneous case. Note that although the firms choose equal quality levels, their qualities are lower compared to the integrated case as α S = < 3 2 k α I = and 2 2 k β S = < 3 2 k β I =. 2 2 k Similarly in the non-integrated case with royalty fees, the analysis closely follows Proof of Proposition. αr S = 55/3 > 2 /3 3 2 k αs = 3 2 k yield qr S = 53 < q S = k 2 Solving the first order conditions for quality decisions simultaneously we get and βs R = 54/3 < 2 0/3 3 2 k βs =. These equilibrium qualities 3 2 k, p SR 3 4 k 2 A = 53 < p S k 2 A =, p SR 3 5 k 2 B = 54 > p S k 2 B =, 3 5 k 2 ( ( S θ R = 5 > θ S =, 2 3 πsr A = 55 > π S k 2 A = 2, and π SR 3 7 k 2 B = 54 < π S k 2 B = k 2 Hence, all of the results presented in Proposition hold in the simultaneous product development case. We provide a numerical analysis of the competition with royalty rate case in Table TA. Table TA: Simultaneous Quality Decision Model Equilibrium Results Integrated Non-integrated No Royalty With Royalty Competition α I = 2.5 α =. α R =.28 α CR =.42 Quality β I = 2.5 β =. β R = 0.94 β CR =.06 q I = 6.25 q =.23 q R =.2 q CR =.5 Price p I = 3.25 p A = p B = 0.4 pr A p R B = 0.2 pcr AH = = 0.5 pcr BH = Demand θ I = 50% θ = 33% θ R = 42% θ = 46% π A = π R A Profit π I = 0.52 π B = π R B π A + π B = 0.83 π R A + πr B = 0.4 πcr AH = 0.5 = πcr B = = 0.95 πcr A Consumer Surplus Social Welfare The numerical values have been calculated assuming k = 0. and α L = π CR B = 0.23 i

3 A2 Asymmetric Cost Parameters The analysis follows the proofs in the text closely. The equilibrium results with asymmetric cost parameters, k A and k B, are given in Table TA2. All the results hold qualitatively under this specification. One interesting finding is that the optimal royalty rate does not depend on the cost parameters; the royalty rate, r, is 3/5 whether costs are symmetric or asymmetric. Thus, in equilibrium, we do not see a subsidization scenario in which royalty fees are negative firm A transfers royalty payments to firm B. Such a transfer would actually be an incentive for firm B to increase quality, but the increase is not large enough to cover the subsidization. Another way to solve both value capture and value creation problem would be to produce and market the complement in house. We explore this option in the next subsection. Table TA2: Analytical Equilibrium Results with Asymmetric Cost Parameters Integrated Non-integrated Royalty=0 With Royalty αi = 2 2 k 2/3 A k/3 B Quality βi = 2 2 k /3 A k2/3 B α = 2 2/3 3 4/3 k 2/3 A k/3 B β = 2 /3 3 5/3 k /3 A k2/3 B αr = 5 5/3 2 3/3 3 4/3 k 2/3 A k/3 B βr = 5 4/3 2 /3 3 5/3 k /3 A k2/3 B q I = 2 4 k A k B q = k A k B q R = k A k B Price p I = 2 5 k A k B p A = p B = k A k B p R A = k A k B p R B = k A k B Demand θ I = 2 θ = 3 θ R = 5 2 π A = k A k B π R A = k A k B Profit π I = 2 6 3k A k B π B = 3 6 k A k B π R B = k A k B 7 π A + π B = k A k B πa R + πr B = k A k B Social Welfare k A k B k A k B k A k B Consumer Surplus 2 7 k A k B k A k B k A k B ii

4 A2. Producing the Complementary Good In-house One solution to the difficulties arising from separate development of the complements would be to produce the other good in-house. This option is superior if one the firms has the technical skills to produce the other good efficiently. For example a firm that has the technology of designing and producing a left shoe also has the technology of designing and producing a right shoe. Therefore, we never see separate firms manufacturing only one of these complements. On the other hand, it is difficult to say the same for another complement pair like a processor and an operating system. Firms can typically achieve high levels of competence at developing and producing one of the goods, but if they also try to master developing and producing the other good they will not be as efficient. For instance, if we assume firm A tries to develop product B in house, it will have a cost parameter k B > k B, which determines the amount of spending for a given quality improvement. Firm A has an incentive to develop good B by itself, if its profit from doing so, π A, is higher than the profit from only producing good A. π A > π A or 2 6 3k A kb > k A k B k B < 5.06k B Thus, it is not profitable for firm A to develop product B by itself, unless its cost parameter for product B is less than 5.06 times the cost parameter of firm B, which specializes in good B. We next investigate firm A s willingness to produce good B in house when it charges royalty fees. Conducting a similar analysis when the A firm can impose a royalty fee, shows that firm A prefers developing product B itself when its cost parameter for developing good B satisfies kb < 3.32k B Thus, compared to the original model, firm A is much less willing to produce good B when it can charge a royalty fee. Said differently, when royalty payments can be imposed, we should expect to see much more specialization in an industry whereby separate firms develop and produce each of the complementary products. For example, consider an industry in which firm A has a cost parameter for developing product B that is 4 times as big as firm B s cost for developing product B ( kb = 4k B. When there are no royalties, firm A wants to develop product B in house and be an integrated firm. By contrast, if it can charge royalty fees, firm A is better-off letting firm B, who is more efficient, develop good B and then extract surplus from firm B through the royalty payments. As an example, in the video game market Sony and Microsoft mostly depend on separate software firms for titles. Only Nintendo s game publishing subsidiary is a main source of hit games for the company s console. It is important to note that Nintendo develops games for its consoles since the 970s. iii

5 A3 The Cost of Quality Has a Variable Component A3. Development and Variable Cost In this case firms incur positive marginal costs in addition to upfront quality investment. While developing a new product R&D cost may increase quite quickly, it is reasonable to assume that variable cost rises more slowly with respect to improvements in quality. Thus, we assume that the marginal cost increases linearly with the product s quality and do not depend on the quantity sold. Firm A pays a manufacturing cost of mα for each product and firm B pays mβ. We find that all of the results hold for low m. We provide a numerical analysis in Table TA3. Table TA3: Development and Variable Cost Model Results Integrated Non-integrated No Royalty With Royalty Competition α I = 2.4 α =.37 α R =.57 α CR =.58 Quality β I = 2.4 β =.9 β R =.04 β CR =.09 Price p I = 2.99 p A q I = 5.76 q =.64 q R =.63 q CR =.72 = 0.57 pr A p B = 0.56 pr B = 0.28 pcr AH = 0.06 = 0.73 pcr BH = 0.90 Demand θ I = 48% θ = 30% θ R = 38% θ = 4% Profit π I = 0.40 π A = π R A π B = π R B = 0.04 πcr AH = 0.09 = πcr B = Consumer Surplus Social Welfare The numerical values have been calculated assuming k = 0., m = 0.05 and α L = 0.25 A3.2 Variable Cost Only We also consider a case in which cost of quality only has a per-product component, i.e. without upfront investment. For tractability this marginal cost should be sufficiently convex and we use 3 cα3 and 3 cβ3. In such a specification, the non-integrated firms, with or without a royalty structure, choose the same quality levels that the integrated firm chooses. Consequently, there is no value-creation problem. This is because margin functions in each model become multiples of each other with constant coefficients once optimal prices are substituted in the profit functions. This is also true for demand functions, hence the first order conditions are maximized at the iv

6 same quality levels in each model. For example, in the integrated case margin is 3α iβi k(α 3 i β3 i 6 and demand is 3α iβi k(α 3 i β3 i 6α i. In the non-integrated case the values are 2 of the integrated case; βi 3 margin is 3α iβi k(α 3 i β3 i and demand is 3α iβi k(α 3 i β3 i 9 9α i. Since we set the first order conditions to βi zero in order to find the optimal quality levels, constant numbers do not change the equilibrium decisions. In the competition model, however, the high-quality A firm has an incentive to set price and the royalty rate such that the low-quality firm has no sales and the win-win-win-win result no longer holds. This is not surprising though, since non-integrated firms already produce the integrated quality level the low-quality firm s role is not needed. A4 Non-strict Complementarity A more general function for the quality of composite product is q = z α + z 2 β + z 3 αβ. In this specification a consumer derives utility from using the two products together as well as from using each product by itself. Note that the main model analyzed in sections 4 and 5 is a special case of this general model with z = z 2 = 0. The general case also nests the non-essential complement case in which q = z α + z 3 αβ (see Chen and Nalebuff We analyze the model with the more general utility function and find that our results continue to hold as long as there is sufficient complementarity, i.e. z 3 is sufficiently large compared to z and z 2. A numerical analysis is given in Table TA4. Table TA4: Non-strict Complementarity Model Results Integrated Non-integrated No Royalty With Royalty Competition αi = 7.82 α = 4.63 αr = 5.35 α CR = 5.38 Quality βi = 7.82 β = 4.07 βr = 3.36 β CR = 3.49 qi = 99. q = 65.2 qr = 62.6 q CR = 65.2 Price p I = p A = 2.74 pr A = 9.82 pcr AH = 4.5 p B = 2.74 pr B = 26.3 pcr BH = 3.52 Demand θ I = 50% θ = 33% θ R = 42% θ = 44% Profit π I = 7.89 π A = 3.94 πa R = 6.02 πcr AH = 6.2 π B = 5.00 πb R = 2.88 πcr B = 3.8 Consumer Surplus Social Welfare The numerical values have been calculated assuming k = 0., z = z 2 =, z 3 = 3 and α L = 0.25 v

7 A5 Enriching the Competitive Setup A5. Firm B Offers Two Versions of Its Product Firm B produces two versions of its product, B H and B L, that are exclusively compatible with A H and A L respectively. Consequently the two product pairs available to consumers are A H B H and A L B L. The qualities of the B products are denoted β H and β L, where β H = γβ L for γ (0, ]. γ is a parameter that captures a possible downgrading of the B product due to compatibility reasons with the lower-quality A product. For instance, in the case of universal apps that run on both the ipad and the iphone, the iphone version needs to be scaled down to be compatible with the iphone s lower resolution. In this case γ would be less than, because the iphone version of the app offers less quality than the ipad version. On the other hand, there are compatible apps that also run on both iphone and ipad, but with the same resolution. For these apps γ = as the quality offered is the same for both versions. Another example follows from the video game industry. A video game title designed for a powerful console like the PS3 may need to be scaled down to run properly on a PC. In some cases the game title is launched later in the PC market in order to allow sufficient time for the average PC to become powerful enough to run the game. In both cases γ is less than because playing a scaled down game or having access to the game several months later compared to the console version decreases consumer utility. We assume that the scaled-down version of the product does not require the B firm to incur further R&D investment. As a result of the need to scale-down B L, and the fact that consumers only care about the utility they derive from the pair, the quality differentiation between product pairs A H B H and A L B L depends not only on the difference between the quality levels of the A products but also on γ; a lower γ results in greater product pair differentiation. All the results continue to hold in this setting. One can define α L = γα L and simply replace α L with α L throughout the analysis to reach the equilibrium quality levels and prices. For instance, the region in Propositions 2 and 3 becomes: α L (0, α L ] γα L (0, α L ] α L (0, α L γ ] and consequently Figure 3 changes such that critical values become α L γ always holds as γ (0, ]. and α L γ. Note that α L < α H A5.2 Endogenous α L In our work we assumed exogenous α L. Finding an analytical expression for optimal α L is not feasible. Instead we will present a numerical example. Note that p AH decreases sharply with vi

8 α L. Actually p AH becomes zero for α L = Thus if r L was endogenous A L could not choose a higher value because in such a case the high-quality firm would start modifying r H to keep A L out of the market. For example as you can see in Table TA5, the optimal α L is around 0.25 for k = 0.. Table TA5: Results of Competition Case with different α L values Royalty Competition α L = 0.05 α L = 0. α L = 0.5 α L = 0.2 α L = 0.25 Quality αr =.68 α CR =.68 α =.68 α =.69 α =.69 βr =.09 β CR =.0 β =.2 β =.3 β =.4 Price AH AL BH BL = pcr AH = 0.24 p AH = 0.64 p AH = 0. p AH = 0.05 = pcr AL = 0.05 p AL = p AL = p AL = = pcr BH = p BH = p BH = p BH = = 0.06 pcr BL = p BL = p BL = p BL = Demand D H = 42% D H = 43% D H = 43% D H = 44% D H = 44% D L = 4% D L = 4% D L = 4% D L = 5% D L = 5% Royalty r H = 0.62 r H = 0.64 r H = 0.65 r H = 0.67 r H = 0.69 Profit π CR AH = 0.58 πcr AH = 0.59 πcr AH = 0.60 πcr AH = 0.60 πcr AH = 0.60 πb CR = πb CR = πb CR = πcr B = πcr B = 0.00 π CR AL = 0.00 πcr AL = πcr AL = πcr AL The numerical values have been calculated assuming k = 0. = πcr AL = A5.3 The Low-quality Firm Also Charges Royalties We consider a scenario in which the low-quality firm also charges a royalty fee. Denote r i as the royalty rate charged by firm A i. For simplicity we assume that r L is exogenous. This would mirror a situation where there is an established platform with a set quality level and royalty rate and a new and higher-quality platform enters the market. The profit functions 7, 8, and 9 become ( πah CR = θ (p AH + r H p BH 3 kα3 H, (A πal CR = ( θ θl (p AL + r L p BL 3 kα3 L, and (A2 ( πb CR = θ ( r H p BH + ( θ θl ( r L p BL 3 kβ3. (A3 vii

9 The analysis closely follows Proof of Proposition 3; hence some details will be omitted. In order to find the sub-game equilibrium we start from the last stage where the firms make their pricing decisions. We derive the first order conditions of the firms profits (given in A, A2, and A3 with respect to the prices p CR AH, pcr AL, pcr BH, and pcr BL. This system yields the following prices: AH = α Hβ[( r H 2 (3 r L α H (3 2r H r L α L] ( r H [(3 r H (3 r L α H α L, ] AL = α Lβ[((2 r H rl 2 (5 3r Hr L +α H ( r L α L] ( r L [(3 r H (3 r L α H α L, ] BH = α H β[( r H (3 r L α H +( r L α L ] ( r H [(3 r H (3 r L α H α L, and ] BL = α Hα L β(4 3r H 2r L +r H r L ( r L [(3 r H (3 r L α H α L. All the second conditions are ] negative and thus the profit functions are strictly concave in prices. Using the prices from the sub-game equilibrium, we revise firm B s profit function in the second stage to πb CR = α2 H β[( r H(3 r L 2 α H +(7 (3 r L r H 6r L +rlα 2 L] 3 kβ3. Firm B selects the profit maximizing quality level β CR = α H Substituting β CR π CR AH = α3 H (α H α L (3 r L 2 [(3 r H (3 r L α H α L ] 2 ( r H (3 r L 2 α H +(7 (3 r L r H 6r L +rlα 2 L k[(3 rh. (3 r L α H α L ] we update the high-quality firm s profit function in the first stage to ( r H (3 r L 2 α H +(7 (3 r L r H 6r L +rlα 2 L k[(3 rh (3 r L α H α L ] 3 3 kα3 H. Now we will find firm A H s optimal royalty rate choice, rh. We differentiate πcr AH with respect to r H, set the first order condition to zero and find the optimal rh = 3(3 r L 2 α 2 H +2(8 7r L+3rLα 2 L 5[(3 r H (3 r L α H α L. All of the second order ] conditions are met. Table TA.6: Results of Competition Case with different r L values Royalty Competition r L = 0 r L = 0. r L = 0.2 r L = 0.3 r L = 0.35 Quality αr =.68 α CR =.69 α =.69 α =.69 α =.69 α =.69 βr =.08 β CR =.4 β =.4 β =.4 β =.4 β =.4 Price p R A p R B = 0.30 pcr AH = 0.05 p AH = p AH = p AH = p AH = AL = p AL = p AL = 0.09 p AL = p AL = 0.00 = 0.75 pcr BH = p BH = p BH = 0.96 p BH = p BH = 0.94 BL = p BL = p BL = p BL =.05 p BL =.2 Demand D H = 42% D H = 46% D H = 44% D H = 44% D H = 44% D H = 44% D L = 3% D L = 5% D L = 6% D L = 6% D L = 7% Royalty r H = 0.69 r H = 0.68 r H = 0.67 r H = 0.67 r H = 0.67 Profit πa R πb R = 0.57 πcr AH = 0.60 πcr AH = 0.59 πcr AH = 0.58 πcr AH = 0.57 = πcr B = πb CR = πcr B = πcr B The numerical values have been calculated assuming k = 0. and α L = 0.25 πcr AH = 0.57 = 0.00 πcr B = 0.00 Our focus is the effect of having the low-quality firm charge royalty profits on our results. Thus, we will not characterize the equilibrium conditions. Instead we will show that all firms are still better-off for some values of r L compared to the royalty case with two non-integrated viii

10 firms in the following table. From Table TA6, it is easy to see that as r L increases, p AL decreases sharply reaching to nearly zero at r L = Thus if r L was endogenous A L could not choose a higher value. A6 Horizontal Differentiation in the A Market In this section we explore the impact of horizontal differentiation in the A market instead of quality differentiation. So we assume exogenous quality level of α for each A firm. The consumers taste parameter is t U[0, ] and the corresponding utility from purchasing product i is U i = θαβ p i p B tx where x is the distance between a consumer s ideal product and the location of firm i. Note that, even though the A firms are not differentiated in quality our specification still has θ, i.e. a consumer s taste for quality still effects her decision to buy and willingness to pay. To keep things simple, we assume two discreet quality levels β H and β L for the B firm. Now we are going to analyze a model in which there is a monopolist A firm located at /2 and a duopoly in the A market located at 0 and. We will characterize the conditions under which the B firm chooses the high quality level when there are two firms in the A market and how this might benefit the A firms. Because we are interested in the impact of competition, we will concentrate on the cases in which the market is covered (t < αβ. When there is a single firm in the A market the demand area consists of two adjacent trapezoids. The edge they share is located at x = /2 and it is ( θ long where θ = p A+p B. The length of the shorter edges are given by ( θ where αβ θ = 2p A+2p B +t. Hence both firms face the same demand 4αβ 4p A 4p B t. From the first order 2αβ 2αβ conditions we find the equilibrium prices p M A π M A = p M B = 4αβ t 2. The profit functions become = (4αβ t2 44αβ 3 kα3 and π M B = (4αβ t2 44αβ 3 kβ3. When there are two firms in the A market, the indifferent consumer along the horizontal dimension is given by x = p B p A+t. Of the consumers located at x, the ones that have θ 2t in excess of of θ = p A+p A2 +2p B +t purchase. Consumers located at x = 0 find product A 2αβ ideal for their (horizontal taste and ( θ of them purchase, where θ = p A+p B. Similarly at x =, ( θ 2 of consumers purchase product A 2, where θ 2 = p A2+p B αβ. Note that αβ if a consumer purchases an A-type product she also buys a B product as well. The demand for A and A 2 is D = (p A2 p A +t(4αβ 3p A p A2 4p B and D 8αβ = (p A p A2 +t(4αβ 3p A2 p A 4p B, respectively 8αβ and the demand for firm B is D + D 2. The equilibrium prices are p D A = pd A2 = 4αβ+9t 97t 2 +8tαβ+6α 2 β 2, p D 8 B = 4αβ t+ 97t 2 +8tαβ+6α 2 β 2 and the corresponding profit functions ( 6 ( 2 are πa D = πd A2 = t 5 97t 2 +8tαβ+6α 2 β 2 4αβ 49t 3 kα3 and πb D = t 97t 2 +8tαβ+6α 2 β 2 +4αβ t 64αβ ix 256αβ

11 3 kβ3. Define γ β H β L. The B firm chooses β L when there is a single A firm and β H = γβ L for two A firms iff. π D B (β H = t ( 97t 2 +8tαβ L γ+6α 2 β 2 L γ2 +4αβ L γ t 2 256αβ L γ 3 kβ3 L γ3 > πb M(β L = (4αβ L t 2 3 kβ3 L, or 44αβ L 737t t 3 αβ L γ+2784t 2 α 2 βl 2 γ2 280tα 3 βl 3 γ3 +256α 4 βl 4 γ4. α 2 βl 8 γ2 (γ 3 2 k < 98t2 8γt 2 296tαβ L γ 28α 2 βl 2 γ+44α2 βl 2 γ αβL 4 γ(γ3 28 Now the only thing remaining is finding out how much quality differential should the available B products have so that the original firm A benefits from competition. (4αβ L γ+49t 5 97t 2 +8tαβ L γ+6α 2 β 2 L γ2 2 The relevant condition is πa D(β H = t 256αβ L γ 3 kα3 > πa M(β L = (4αβ L t 2 44αβ L 3 kα3, or γ < 45 97t 8 688t 7 αβ L +2400t 6 α 2 βl 2 008t5 α 3 βl t4 α 4 β 4 9(49t L 4 76t 3 αβ L +784t 2 α 2 βl 2 4 (t 4 +2t 3 αβ L 992t 2 α 2 βl 2 +32tα3 βl α4 βl 4 2 4((t 4 +2t 3 αβ L 992t 2 α 2 βl 2 +32tα3 βl α4 βl. 4 In other words as long as there is sufficient quality differential between available B designs and the cost parameter is low such that the B firm chooses the high-quality product when there are two A firms but not when there is a single A firm in the market, competition is beneficial for A firms. The consumer taste parameter, t, needs to be moderate for this result to hold. It needs to be low enough to induce price competition between the A firms in favor of firm B, but it also needs to be high enough so that the loss from price competition is confined. In order to make this result easier to interpret we provide a numerical result: For α =, t = and β L =, the higher-quality B product should satisfy β H >.69 or equivalently, γ < Let γ < 0.5, then the cost parameter should satisfy k < 0.55, so that the B firm finds choosing the higher quality level worthwhile. Note that this is not a stringent condition, in other numerical examples we have regularly used k = 0.. This is not surprising though, because the existence of two firms in the A market gives a lot of incentive to the B firm to increase quality. As the B firm can enjoy higher margins due to competition in the A market and also enjoys larger demand because two A firms cover a larger part of the horizontal dimension, only an exceptionally high cost parameter can deter it from choosing higher quality. A7 Other Contractual Agreements Between the Firms A7. Non-linear Royalty Rate We first consider the case where the level of the royalty rate that firm A charges firm B is a function of p B. The first order condition for p B is αβ [( r(αβ p A 2p B r (αβ p A p B p B ]. Note that the non-integrated case with royalties presented in the paper is nested in this specification, where r = 0 and αβ p A 2p B = 0. For price coordination, we know that the optimal p A + p B has to decrease. For small decreases (in prices the first term in the x

12 first order condition becomes positive and the second term has to be negative to balance this. As αβ p A p B is positive it follows that r is positive as well. Thus, the optimal royalty rate that achieves price coordination increases with p B. We now show that this royalty function cannot achieve quality coordination since firm B s quality decreases with the royalty rate. The first order condition of firm B s profit function with respect to quality is: = π B + β π B dp B + π B dp A + π B r dp B. We evaluate this expression at β = β(r dβ p A dβ r dβ L where r L is the optimal royalty rate in the non-integrated case with royalties (a constant and p A and p B are the optimal prices given β = β(r L. can see that π B r r rates, π B β r dp B is negative because π B dβ r > 0 as we have shown above and dp B dβ = p B(p B +p A ( r αβ 2 π B dπ B dβ As these are the optimal prices, = π A = 0. We < 0 as firm B s profit shrinks with royalty > 0 as price increases with quality. Note that 3k B β 2. This expression equal zero at β = β(r L with the p A and p B being the non-coordinated optimal prices. Because the non-coordinated prices are lower than the optimal prices, the term p B(p B +p A ( r is lower at the coordinated prices than the same term αβ 2 at the non-coordinated price, while 3k B β 2 remains the same and therefore π B (β = β(r β L at the optimal prices is negative. Since all the terms of the first order condition are non-positive (either zero or negative, coordinating the price reduces firm B s quality. A7.2 Two Part Tariff: Fixed Fees Plus Royalty The second contractual agreement extension is a two part tariff in which Firm A charges firm B a fixed fee plus royalty payments per each unit sold. The difference between this model and the model we analyze in the paper is the addition of the fixed fee. Obviously, the contract we have analyzed in the paper is nested within this contract. However, we will show that even this contract does not fully coordinate the system. In the two part tariff case firm A first decides on quality and the terms of the contract before firm B decides on quality level. Firm A s profit function is linearly additively separable in the fixed fees. Therefore the optimal fixed fee is a corner solution and firm A can extract all of firm B s profit. Thus, F = 2[α( r] 3 2 3(3 r 3 k B. Using this we solve for the optimal qualities and prices and compare the solution to the coordinated case. We find that the two part tariff contract does not achieve full coordination and the profits are lower. A7.3 Royalty Payment which is Fixed per Unit In the last contractual agreement we consider the royalty is a fixed fee per unit sold instead of a percentage of the sales. The firms profit functions in this case are given by: xi

13 ( π A = ( (p A + f 3 kα3 and π B = (p B f 3 kβ3, where f denotes p A+p B p A+p B αβ αβ the per unit royalty fee. From the first order conditions in the pricing stage, it is easy to see that the optimal price that firm A charges is p A f and the optimal price that firm B charges is p B + f, where p A and p B are the optimal prices in the non-integrated case without royalty fees. Therefore the total each firm makes per unit is equal to what they would make in the nonintegrated case without royalties, e.g. firm A earns p A f + f = p A. In other words charging fixed royalty payments per unit brings no improvement at all. xii