Setting Single-Period Optimal Capacity Levels and Prices for Substitutable Products John R. Birge John Drogosz Izak Duenyas Department of Industrial & Operations Engineering University of Michigan Ann Arbor, MI 48109 Technical Report 97-03 April 1997

Setting Single-Period Optimal Capacity Levels and Prices for Substitutable Products John R Birge, John Drogosz and Izak Duenyas Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109 April 1997

Setting Single-Period Optimal Capacity Levels and PIices for Substitutable Products John R. Birge, John Drogosz, and Izak Duenyas Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI, 48109 Abstract In this paper, we consider the determination of optimal capacity levels and prices for two substitutable products in a single-period problem. We first consider the case where the firm is a price taker but can determine optimal capacity levels for both products. We then consider the case where the firm can set the price for one product and the optimal capacity level for the other. Next, we consider the case where capacity is fixed for both products, but the firm can set prices. For each of these cases, we examine the sensitivity of optimal prices and capacities to problem parameters. Finally, we consider the case where each product is managed by a product manager trying to maximize individual product profits rather than overall firm profits and analyze how optimal price and capacity decisions are affected. 1 Introduction Firms must continuously make pricing and capacity decisions to respond to market forces. Facing uncertain demand, firms must balance pricing and production decisions to respond to market demand. Many firms produce a variety of products, some of which may be substitutable by consumers. The fact that products are substitutable makes pricing and capacity decisions more difficult. This is due to the fact that the firm needs to consider 1

the effect that a change in the price of one product is going to have on the demand level for another product. A good example is in electronics manufacturing where a firm might produce a variety of chips. The price of a faster chip affects the demand for a slower chip as well since if the prices for the two types of chips are sufficiently close, many customers might opt for the faster chip, thereby significantly decreasing demand for the slower chip. In this paper, we study single-period pricing and capacity setting decisions for a firm that produces two substitutable products. Our aim is to build intuition on how pricing and capacity decisions change as a function of costs, demand functions, and preexisting capacity levels; we therefore focus on simple single-period models. This also enables us to contrast our results with the famous news vendor problem which is a single period problem where the capacity (production level) for only a single product is chosen. We also focus on the issue of centralized versus decentralized decision making and how this affects the nature of the decisions made. Many firms assign a product manager to each product that the firm produces, and the role of the product manager is to maximize the profits made by the product assigned to the manager. If all products are produced on separate production lines, and the products are unsubstitutable, the decisions made by each product manager trying to maximize product profits will correspond to decisions made by a centralized controller maximizing system-wide profits. The more interesting case is when products are substitutable; in that case, we analyze when the system-optimal decisions are the same as the individual product-optimal decisions. The classic problem where price is known and capacity is uncertain is the news vendor problem. Following this classic problem, many extensions have been made. For example, Ismail and Louderback [8], Lau [11] and Kabak and Schiff [9] have studied a one product news vendor problem where the probability of achieving a pre-determined profit level is maximized. Ismail and Louderback[l], and Sankarasubramanian and Kumaraswamy [14] and Lau and Lau [12] have studied a single product news vendor problem where the demand level is dependent on the price set. Therefore, both optimal price and optimal order quantity are determined. Li et al. [13] focus on a two-product news vendor problem where the 2

probability of achieving a profit target is maximized. There have been a variety of attempts to introduce the effects of capacity constraints on the price and production decisions in the news vendor problem. Kreps and Scheinkman [10] examine a problem where two identical firms compete with products that are perfect substitutes. The demand for each product is a function of its own price. The two firms enter into a two-stage competition where they set capacities in the first stage and make pricing decisions independently in the second stage. Staiger and Wolak [15] focus on two firms who produce the same product and have the same capacity costs. They analyze an infinitely repeated game where prices are adjusted periodically, and examine the effects of having excess capacity. Examples of research involving substitutable products includes Ignall and Veinott [7] who examine the optimality of myopic policies with several products. Bassok et al. [1], Bitran and Dasu [2], Hsu and Bassok [6], and Gerchak et al. [5] study ordering policies with substitutable products while Carmon and Nahmias [4] examine lot-sizing decisions in semiconductor manufacturing where the products are substitutable. In this line of research, it is assumed that the manufacturer is a price taker for all of its products. We study the case of a firm producing two products with price dependent demands where the firm has the ability to make pricing and/or capacity decisions for one or both of its products. We begin in section two by discussing the case where a firm is a price-taker for both of its products but has control over the amount of capacity to install for each product. In section three, we analyze the situation of a firm which needs to decide on the amount of capacity to install for one of its products and the price to set for the other product. This is a situation facing firms who introduce a new and improved product which has the potential to cannibalize sales from its existing product and where the firm only has a limited amount of capacity online for this new product. An example of this would be a microchip manufacturer launching the next generation of microchips after its existing product has been cloned. In section four, we examine the case where both products have a given capacity constraint and the firm sets prices for both products. This situation is 3

commonplace for firms who have no ability to increase capacity in the near term but can only control sales and profits by adjusting their prices. We end in section five with a discussion on how the decisions made in the previous sections differ when decisions regarding price and/or capacity are made sequentially instead of simultaneously. These situations can arise when a firm has different brand managers making decisions to maximize each product's profit independently rather than maximizing system wide profits. We use the following notation throughout the paper: Pa: price of product A; Pb: price of product B; Ca: production capacity for product A; Cb: production capacity for product B; qa: per unit variable cost of product A; qb: per unit variable cost of product B; ia: cost of adding one unit of dedicated capacity for product A; ib: cost of adding one unit of dedicated capacity for product B; ua(Pa, Pb): mean demand for product A; ub(Pa, Pb): mean demand for product B. For analytical simplicity, we assume throughout the paper that: (Al) the demands for product A and B are distributed uniformly over the intervals [ua(Pa, Pb)r, ua(Pa, Pb) + r] and [ub(Pa, Pb) - S, ub(Pa, Pb) + s] where r and s are the ranges of realizable demands above/below their respective means. We assume further that: (A2) the unit investment costs plus variable production costs do not exceed the product price; ia + qa < Pa and ib + qb < Pb with ia, qa, ib, qb > 0; 4

(A3) the mean demand of product A, ua(Pa, Pb), is decreasing in Pa and increasing in Pb and the mean demand of product B is increasing in Pa and decreasing in Pb. Note that (A3) immediately follows from the fact that the products are substitutable. 2 Capacity Decisions for a Price Taking Firm We begin our discussion with a firm that manufactures two products and must decide on the amount of capacity to install to manufacture each product. In this situation the firm is a price-taker in both of the markets in which it competes. For this situation, Lau and Lau [7] provide a solution procedure for determining the optimal capacities to achieve a given probability of obtaining a profit target. In our model, we maximize the expected profit and derive the sensitivity of the optimal capacities to changes in key parameters. The firm's profit function is defined as: Ca R(Ca,Cb) = (Pa - a) X/ x fa(Pa, Pb, a)dXa (2.1) Ua(Pa,Pb)-r ua(Pa,Pb)+r +(Pa - qa) Ca fa(Pa, Pb, Xa)dXa Ca Cb +(Pb - qb) / Xb - fb(Pa, Pb, Xb)dxb Ub(Pa,Pb)-S Ua(Pa,Pb)+s +(Pb - qb) 'Cb J fb(Pa, Pb, Xb)dxb Cb -ia ' Ca - ib Cb, where fa(Pa, Pb, xa) = and Xa is the demand for product A and fb(Pa, Pb, xb) = and xb is the demand for product B. We first show that a unique set of capacities exists that maximizes (2.1). 5

Proposition 1 If Pa > qa > 0 and Pb > qb > 0, then there exists a unique maximum of (2.1) over Ca > 0 and Cb > O. The optimal capacities are: Ca = (ua(Pa, Pb) + r) - ia and Cb = (Ub(Pa, Pb) + s) 2-qb. Proof: To ensure that a unique maximum exists for positive values of Ca and Cb, a solution must exist to the first order conditions and all second order conditions must be satisfied. The first order conditions are: OR(CaCb) = _i _ Ca-(Pa-qa) + (ua(Pa,Pb)+r)-(Pa-qa) = 9Ca - a 2r 2r - aR(Ca,Cb) _ Cb(Pb-qb) + (b(Pa,Pb)+s)-(Pb-qb) -. aCb -ib 2s 2s Rearranging the terms above, we obtain: Ca = (ua(Pa, Pb) + r) - 2P- > 2r(l - ( a ) and (2.2) (Pa - Qa) (Pa - qa) 2. ib' s ib Cb = (Ub(Pa, Pb) + s) - (b ) > 2s(1 - -b)) (2.3) (Pb - qb) (Pb - qb) which are positive by assumption (A2) that some profit can be earned on each product. The sufficient conditions for the existence of a maximum are: 02R(Ca, Cb) (Pa - qa) - 0 (2.4) OC ~ 2r and O2R(Ca, Cb) O2R(Ca, Cb) 9 2R(Ca, Cb) _ (Pa - qa)(Pb - qb) ac2 0C02 (~aO~b 4rs> 0, (2.5) Ca 9b ~ 9caCb 4rs which both hold by assumption (A2), completing the proof. O We now turn our attention to how the optimal capacity decisions vary with changes in the key parameters of the model. 6

Proposition 2 The changes in the optimal capacity mix to changes ines the parameters are: (a) if Pa increases, then Ca decreases, if aPa | > (a -qa)2 and Cb increases; (b) if Pb increases, then Ca increases and Cb decreases ifUb(PaPb > 2itb' 1OPb (Pb-qb) (c) if qa increases, then Ca decreases and Cb does not change; (d) if qb increases, then Ca does not change and Cb decreases; (e) if ia increases, then Ca decreases and Cb does not change; (f) if ib increases, then Ca does not change and Cb decreases; (g) if r increases, then Ca increases, if (Paqa)2i > 0, and Cb does not change; (h) if s increases, then Ca does not change and Cb increases if (Pb-qb) 2b > 0. Proof: We show below the methodology for (a). The proofs of the other cases are similar. From Equation 2.2, we know that: Ca = (ua(Pa, Pb) + r) - i, Taking the derivative with respect to Pa, we obtain aC &lua(Pa,Pb) 2-ia-r P ~ 9Pa +o (Pa-qa )2 ' Since a(PaPb) < 0 by assumption(A3), we therefore conclude that Ca will decrease if OPPa a ua(PaPb) > 2p a-)2 when Pa increases. aPa (Pa,-qa)2 Similarly, from Equation 2.3, we know that Cb = (u b(Pa, Pb) +) - '-s and thus = b(P b) > 0 by assumption (A3). Therefore, we conclude that Cb will always increase as Pa increases. O Parts (a) and (b) of Proposition 2 point to an interesting phenomenon; namely that an expectation of a price increase in a product which will result in a decrease in the mean demand for that product does not necessarily result in the firm decreasing the available capacity for that product. Essentially, the condition in part (a) shows that if the mean decrease in demand is not fast enough and the cost of capacity is not too expensive, it may in some cases be more profitable to increase capacity for that product. This is because even though on average there is less demand for that product, the unit profit may be higher and therefore more capacity might be profitable. The following example shows the possible 7

behaviors for Ca mentioned above. Example 1: Case (i): Ca and Cb both increase with an increase in Pa. In this case, let qa = 3, qb = 2, C= 2000, ci = 60, c2 = 50, r= 400, D= 3000, di = 100, d2 = 19, s= 250, ia = 1, ib = 1, Pa = 6, and Pb = 10. Here, the optimal solution is Ca = 2273.33, Cb = 2301.5, and R(Pa, Pb) = 18592.58. If we increase Pa from 6 to 7, we observe that Ca increases to 2280, Cb increases to 2320.5, and R(Pa, Cb) increases to 20652.25. Case (ii): Ca decreases and Cb increases with an increase in Pa. For this example, suppose qa = 3, qb = 2, C= 2000, c1 = 60, c2 = 50, r= 400, D= 3000, di = 100, d2 = 19, s= 250, ia = 1, ib = 1, Pa = 10, and Pb = 10. In this case, the optimal solution is Ca = 2185.71, Cb = 2377.5 and R(Pa, Pb) = 26168.39. By increasing Pa to 11, we find that the optimal Ca decreases to 2140, the optimal Cb increases to 2396.5 and R(Pa, Pb) increases to 27774.25. OJ. Similarly, parts (g) and (h) of Proposition 2 show that an increase in the variability of demand for a product may result in an increase or a decrease of the capacity for that product. However, changes in variable production or capacity costs always have monotonic consequences on optimal capacity levels. 3 Setting the Price of A and the Capacity Level of B In this section we suppose that a firm currently has fixed (limited) capacity for a new product it is introducing but must decide how much capacity to maintain for its existing product. For example, in electronics manufacturing, when a firm first introduces a new product, the capacity is extremely limited due to low yields for the new product and the necessary time for the factory to ramp up for production (e.g., building a new faster chip). On the other hand, the older product that the firm produces may already be a stable product in a market with active competition. For the older product, the firm might have the option to change its capacity; however, it is a price taker for it. Therefore, the firm 8

faces the joint problem of setting a capacity level for one product and a price level for the other. In this situation, the firm's profit function is defined as: R(Pa, Cb) Ca (Pa-qa) fJ Ua (Pa,Pb)-r Xa fa(Pa, Pb, a)da (3.6) +(Pa - qa) * Ca +(Pb - qb) ' Ub(Pa, +(Pb-qb) - Cb Ua(Pa J Cb Pb)-s Ua(Pa, Cb,,Pb)+r / fa(PaPb, xa)da a Xb ' fb(Pa, Pb, Xb)dXb,Pb)+S fb(Pa, Pb, Xb)dXb - ib Cb. In this section and we assume that: the next, we assume linear mean demand functions. In particular, (A4) the mean demand of product A, Ua(Pa, Pb) = C - ClPa + C2Pb > 0 and Ub(Pa, Pb) D - dPa + d2Pb > 0. The use of such linear functions to model product demands is widespread in the economics literature (see Bulow [3] and Stokey [16]). We also assume that: (A5) c1 > c2 > 0, di > d2 > 0, ci > d2 > 0, and d1 > 2 > 0. Note that these assumptions are reasonable for the following reasons: * the assumption that cl > c2 and d1 > d2 means that the demand function for product A is more sensitive to changes in the price of product A than changes in the price of product B (and similarly for the demand function for product B), and * the assumption that c\ > d2 (and d1 > c2) ensures that an increase in the price of a product cannot increase overall demand, since for example increasing the price of 9

product A would decrease the mean demand for A more than it would increase the mean demand for B (i.e., one cannot create extra demand by increasing the prices of both products). Finally, (A6) for products whose capacity cannot change, we assume that their capacity is above the minimum realizable demand. Proposition 3 If ua(Pa, Pb) and ub(Pa, Pb) are both linear in Pa and Pb and the capacity Ca for product a exceeds the lower bound on demand, ua(Pa,Pb) - r, then the function R(Pa, Cb) achieves a unique maximum over Pa > qa > 0 and Pb > qb > 0. Proof: It is sufficient as in Proposition 1 to show existence of a solution to the first order conditions and to show that the Hessian, H, of R(Pa Cb) is negative definite over Pa > qa > 0 and Pb > qb > 0. The first order conditions are: aR(Pa, Cb) [C2 - (ua(Pa, Pb)- r)2] Ca [ua(Pa Pb) + r - Ca] aPa 2r 2r +(Pa - qa) [Ca-(ua(Pa, Pb) -r)] 9ua(Pa, Pb) 2r ' Pa (Pb- qb) [Cb- (ub(Pa, Pb)- z)] aub(Pa, Pb) 2z OPa =, and R(Pa, Cb) -i + (ub(Pa, Pb) + s - Cb) (Pb- qb) (3. 0Gb + 0. --- = ~- (3.8) 9Cb 2s Solving the above equations, we can obtain the optimal values for Pa and Cb. For the second order conditions, the terms of H are: 82R(PaCb) [Ca-(U(PaPb)-r)] uaPab) HI, = -Pa- = r * a-a 10

(Pa-a) [9UP - ( P-(a(P, Pb)2-r)). a u(PPb) (P [(&bPPb2 - (a - (u(Pa, Pb) - 2)) 2 Ub(a, 2 2 (Pb-qb) Ub6a6A) 9Ub(Pa,Pb) H21 = H12 = aPaaICb 2p aPa _ a-pqca ) [ _(P,H2=HlpC= )_ (Pb-qb) &b(Pa,Pb H22 - - 2s P Cb The sufficient conditions for concavity are: H11 < 0, H22 < 0, (3.9) and H11H22 - H12 > 0. (3.10) For H11, notice that a(PPb) < 0 by assumption (A3) and the capacity of A is greater than the minimum realizable demand by (A6). The first term is, therefore, negative. The second derivative elements in the next two terms of H11 vanish so that the two terms both become negative and yield H11 < 0. For H22, negativity follows by assumption (A2). Inequality (3.10) follows by first noting that the (( b). au),Pb)2 terms from H11H22 and H22 cancel. The remaining terms are Hc H H 2 ( t[Ca - (Ua(Pa, Pb) - r)] dua(Pa, Pb) (Pa - qa) (ua(Pa, Pb)2\ (Pb - qb) 1122 12 = r aPa 2r aa 1 J 2s which is the product of two negatives by our previous arguments in the proof of (3.9). The result follows. O Once again, we are interested in how the optimal decisions change as a function of the optimal parameters. Proposition 4 Assuming the conditions in Proposition 3, (a) if qa increases, then Pa increases and Cb increases; (b) if qb increases, then Pa decreases and Cb decreases; (c) if ib increases, then Pa decreases and Cb decreases; 11

(d) if C increases, then Pa increases and Cb increases; (e) if cl increases, then Pa decreases and Cb decreases; (f) if c2 increases, then Pa increases and Cb increases; (g) if D increases, then Pa does not change and Cb increases; (h) if dl increases, then Pa does not change and Cb decreases; (i) if d2 increases, then Pa increases and Cb increases; (j) if Ca increases, Pa and Cb decrease. Proof: The goal here is to identify the effect of changes in one parameter on the optimal solution of the first order conditions in (3.7) and (3.8). To reflect changes in parameters other than the decision variables, we add extra terms to the definition of R so that R(Pa, Cb) is also written as R(Pa, Cb, qa) when we explicitly consider changes in qa. In this case, the first order conditions in (3.7) and (3.8) are Vcb,paR(Pa, Cb, qa)= 0, where VCb,Pa refers to the partial derivatives with respect to Cb and Pa alone. We suppose the solution to (3.7) and (3.8) is (C*, Pa*) when qa = qa In the following, we use the notation V2 /Z for the partial differential operator given by (2 \ ~yaz 02 acz We next consider changes from q* to qa = qa + bqa. The solutions of (3.7) and (3.8) are then Pa = Pa* + bPa and Cb = Cb + 6Cb- We wish to find the sign of 6Pa and 6cb given 6q > 0. First, using the first order properties, implicit function theorem, and that VCb,PaR(Pa, C, qa) = 0, we must have 0 = Vcb,paR(Pa,Cb, qa) - VCb,pa/R(P Cb, qa)P + VCba/CbR(Pa Cb, qa)Cb + VCb,Pa/qaR(Pa, CX q )6q +EPa6Pa + ECb6Cb + Eqa6qa, 12

where Epc, ECb, and Eq. all approach zero as 6qa approaches zero. For small changes 6q, we therefore seek 6Pa and Scb to solve VCb,Pa/PaR(Pa* Cb, q*)6Pa + VCb,Pa/CbR(Pa*7 Cb, q )D Cb = -VCb,Pa/qaR(P Cb q*)bqa. To simplify the notation, let: Hq = [V,p R(Pa, Cb, qa)] 82R(Pa,Cb) a2R(Pa,Cb) aP aoPa8Cb a2 R(Pa,Cb) a2R(Pa,Cb) Pa Cb Cb It is straightforward to show that the determinant of the matrix Hqa has the same sign as the Hessian of R(Pa, Cb) in the proof of Proposition 3 since the only addition has been 6 to the Pa terms of that matrix. Thus, we have 92R(PaCb) 2R(PaCb) det qa QPa9qa PCb 2 R(Pa,Cb) 2 R(Pa,Cb) = qa aCbqa a Pa b det Hqa > 0 2r Ca [C-Ua(aPb)-r) au(Pab)] (Pb-b) Ub(Pa,Pb) det a ( —2r[- [ (P 2s aP >0 0o,(Pb-Q) 0 -(pbl2s det Hqa > 0 Since a9a(PaPb) < 0, the numerator above will always be positive. Therefore, as qa increases, Pa increases. The change in Cb is a92R(PaCb) 92 R(Pa,Cb) de2 — q, aP.aqa a2R(Pa,Cb) _ aR(PCb) det qdtPaCb qa > Cb9qa 6Cb detJ, > det Hqa > 0 13

[Ca-(ua(Pa,Pb)-r)] a'U(Pa,Pb) r ' P (Pa-qa) (9Ua(PaPb)\2 [Ca-(Ua(PaPb)-r)] ua(Pab)] det 2r O [ 2r a2 det Hqa > 0 Since ua(P Pb) < 0 and au(pa,P) > 0, the numerator above is again positive by assumption (A3). Therefore, as qa increases, Cb increases. Repeating the procedure above, we arrive at the other conclusions for (b) through (i) in Proposition 4. (j) In the cae where we increase Ca, we have four potential cases: (i) Pa and Cb increase, (ii) Pa increases and Cb decreases, (iii) Pa decreases and Cb increases, (iv) Pa and Cb decrease. Cases (i) and (ii) are not possible, since given the added capacity, one would never increase price. If Pa increases, then demand will decrease for product A and increase for product B. If this strategy improved profits in case (i), it would have been done at the lower level of capacity for product A. In case (ii) if the price of A increased, we would not decrease the capacity of product B since it would be in greater demand. We may also discard (iii) as a plausible outcome given that when Pa decreases demand for product B will also decrease. Hence, we would not want to increase the capacity of product B. If this strategy would improve the overall profits, it would have been implemented under the initial capacity level of A. Therefore, we are left with the conclusion that as Ca increases, Pa and Cb decrease. Example 2: When Pb is changed, we may observe different behaviors in the decision variables. We now provide examples of the three possible behaviors that can occur when we increase Pb. Case (i): Pa and Cb increase In this case let Ca = 500, qa = 2, qb = 2, C= 2000, ci = 100, c2 = 99, r= 420, D= 3000, di = 100, d2 = 10, s= 1000, ib = 1 and Pb = 5. Here, the optimal solution is Pa = 18.24, 14

Cb = 3015.8 and R(Pa, Pb) = 12218.8. If we increase Pb from 5 to 6, we observe that Pa increases to 19.16,Cb increases to 3091.628 and R(Pa, Cb) increases to 15006.61. Case (ii): Pa increases and Cb decreases For this example suppose Ca = 500, qa = 2, qb = 2, C= 2000, c1 = 350, c2 = 18, r= 420, D= 3000, di = 100, d2 = 99, s= 1000, ib = 1 and Pb = 15. We obtain an optimal solution of Pa = 5.29, Cb = 2869.44 and R(Pa, Cb) = 24510.26. By increasing Pb to 16, we find that the optimal Pa increases to 5.34, the optimal Cb decreases to 2785.52 and R(Pa, Cb) increases to 25312.3. Case (iii): Pa decreases and Cb increases In this example let: Ca = 500, qa = 2, qb = 2, C= 2000, c1 = 350, c2 = 99, r= 420, D= 3000, d1 = 100, d2 = 5, s= 1000, ib = 1 and Pb = 3. The optimal solution is Pa = 5.36, Cb = 1726.814 and R(Pa, Cb) = 1181.003 By increasing Pb to 3.3, we see that Pa decreases to 5, Cb increases to 2156.558 and R(Pa, Cb) increases to 1867.28. E. In Proposition 4, we discover that, when a firm has control over the price of one of its products and the capacity level of the other, changes in the production costs of either product have an impact on both decision variables. If the production cost of the product which we have price control over increases, then we should raise the price on this product and also increase the capacity of the other product. On the other hand, if the production cost increases on the product whose price we cannot control, it is best to decrease its capacity and to increase the price of the other product. We find that if the capacity investment costs for product B increase, we should increase our price on product A and reduce the capacity for product B. We also discover that we will increase the price of product A and the capacity of product B if the mean demand for product A increases by a constant (i.e., the mean demand shifts upwards). However, if the mean demand for product B changes in the same fashion, we will keep the price of product A the same and increase the capacity 15

for product B. In the case where the price elasticity for product A(cl) increases, we will decrease both the price of product A and the capacity level for product B. When the price elasticity for product B increases however, we will not change the price of product A but we will decrease the capacity for product B. If the cross price elasticities increase, we should increase both the price of product A and the capacity for product B. When the capacity for product A is increased, then the price of product A and the capacity for product B will decrease. We also observe that when the price of the product in the market in which a firm is a price taker increases, the firm's optimal strategy will differ depending on the price elasticities of the firm's products. 4 Pricing Decisions: Two Products with Capacity Constraints We now examine the case where both products being manufactured have a given capacity constraint and the firm must set prices for both of its products. The situation modeled here reflects the situation where the high investment costs for new capacity make it unprofitable to build new capacity (e.g., because it is forecast that the products are at the end of their lifecycles and therefore it is not possible to recoup the costs of new investment) and therefore the firm can only exercise pricing control. We will again assume that the demand for products A and B are distributed uniformly and that the mean demands are linear in prices. The profit function is now defined as: Ca R(Pa,Pb) = (Pa-qa)- J Xa fa(Pa, Pb, a)dXa (4.11) Ua (Pa,Pb )-r ua(Pa,Pb)+r +(Pa - qa) * Ca J fa(Pa, Pb, Xa)dXa Ca Cb +(Pb - qb) * b * fb(Pa, Pb, b)dXb Ub(Pa,Pb)-s 16

+(Pb-qb) ' Cb Ua(Pa,Pb)+8 I fb(Pa,Pb, b)dxb Cb Proposition 5 Under assumptions (Al) through (A6), the function R(Pa, Pb) is concave in Pa and Pb. Proof: The first order conditions for optimality are: aR(Pa, Pb) aPa [C- (ua(Pa, Pb) - r)2] Ca [ua(Pa, Pb) + r - Ca] 2r 2r (Pa - qa) [C - (ua(Pa, Pb) -r)] 9ua(Pa, Pb) 2r aPa (Pb- qb) [Cb- (ub(Pa, Pb) -s)] ub(Pa, Pb) 2s aPa = 0 [C2 - (ub(Pa, Pb)- s)2 Cb [Ub(Pa, Pb) + s - Cb] 2s 2s (Pb - qb) [Cb - (ub(Pa, Pb)- s)] aub(Pa, Pb) 2s aPb (Pa - qa) [Ca - (ua(Pa, Pb) - r)] ua(Pa, Pb) 2r aPb 0 (4.12) a2R(Pa, Pb) oPb (4.13) The terms of the Hessian of the profit function are: - 22R(Pa,Pb) [Ca-(Ua(Pa,Pb)-r)] aUa(Pa,Pb) H11 ap- p2 r- (P-2r-a [( aPaP )2 (Ca - (ua(Pa, Pb) - r)) - 2Ua(PaPb) -(Pb -q) [(b(PaPb )2 (Cb- (Ub(Pa, Pb) - s)) *2b(PaPb) 2 L a ] 2 R(Pa,Pb) [Ca-(Ua(Pa,Pb)-r)] Ua(Pa,P) [Cb-(Ub(Pa,Pb)-S)] oUb(Pa,Pb) H21 - H12- 9PaOPb - 2r- Pb + 2s aP (Pa -a) [Ua(Pa,Pb) 9Ua(Pa,Pb) (Ca a(Pa Pb)- r)) 2 Ua (Pa,Pb)] _ 2 [ (Pa-,P P -,. PaPb (Pb-qb) 9Ub(PaPb) OUb(Pab) 92Ub(Pa,Pb) 2r (Ca - (Ub(Pa, Pb) - r0)). 49Pa OPb ' OPa aPb H _ 2R(Pa,Pb) _ [Cb-(ub(Pa,Pb)-s)] ub(Pa,Pb) H22 - s Pb (Pb-qb) [(ub(P ),Pb) 2 (CG- (ub(Pa, Pb) s)) a2Ub(Pa,Pb 1 2s OPb -op-t ) 2~~~~~~~~~~~~~~ 17

22 (Pa -qa) [(a(gPb) _ (Ca- (ua(Pa Pb) r)). a(PaPb) 2r (C UPPb ba aa J) p2 The sufficient conditions for concavity are: H11 < 0, H22 <0, (4.14) and HllH22- H12 > 0. (4.15) For H11 and H22, notice that aU(PaPb) < 0 and 'Ub(PaPb) < 0 by assumption (A3) and Ca and Cb are greater than their respective minimum realizable demands by assumption (A6). The first terms of H11 and H22 are, therefore, negative. The second derivative elements in H11 and H22 vanish so that the last two terms of both become negative and thus yield H11 < 0 and H22 < 0. When we calculate the determinant of the Hessian and simplify, we obtain the following expression (Ca-(aa (PaPb)-r)(Cb-(ub(Pa,Pb)-s)(cldl -c2d2) (Ca-(Ua(Pa,Pb)-r)(Pb-qb)(cl d-c2dd2 ) 2rs 2rs (Cb-(Ub(Pa,Pb)-s)(Pa-qa)(cdl-ClC2d2) ) (Pa-qa)(Pb-qb)(cld -C1C2dld2) 2rs 2rs + (Pa-qa)(Pb -qb)d2 > O 4rs 2 0 We know from assumption (A5) that ci > c2, dl > d2 and from assumption (A6) the capacities for each product will be greater than or equal to the lower bounds on their respective demands. In addition, as mentioned in Section 1, we assume that the price of each product is greater than the sum of the variable production costs and investment costs (A2). Consequently, all of the terms in the expression above will also be positive and thus the determinant of the Hessian is positive. O Proposition 6 Under the assumptions of Proposition 5, (a) if C increases, then Pa increases, Pb increases; (b) if cl increases, then Pa decreases, Pb decreases; 18

(c) if c2 increases, then Pa increases, Pb increases. Given the symmetry of the problem, the parameters D, d1 and d2 will exhibit the same behavior as their counterparts above. Proof: (a) Following the same methodology as in Section 3, we will now show the case where the demand parameter C is increased from C to C + 6. The proofs of the other parameters are done similarly. We again have two linear demand curves: ua(Pa, Pb) = C - cPa + c2Pb and ub(Pa, Pb) = D - dlPb + d2Pa. We define Hc as: a2 R(Pa,Pb ) a2 R(P,Pb) OP2 SPaP9Pb Hc= aPa a2R(Pa,Pb) 0(Pa,Pfb) aPa 09 ap,2 Loeaob It is straightforward to show that the determinant of the matrix HC has the same sign as the Hessian of R(Pa, Pb) in the proof of Proposition 5 so that det Hc > 0. We now examine the changes in the variables to an increase in the parameter C. The change in the price of product A when C increases is: (Ca-(a(Pq,Pb)-r))C2 + (Cb-(ub(Pa,Pb)-s))d2 R R(PaPb) 2r 2s det 2a+(Pa-qa)cIC2 + (Pb-qb)dld2 + 2r + 2s _ 2R(Pa,Pb) (Cb-(Ub(Pa,Pb) —))dl _ (Pa-qa)c _ (Pb-qb)d P_ aPbaC s 2r 2s bpa det He > 0 (Ca-(ua(Pa,Pb)-r))c2 + (Cb-(Ub(Pa,Pb)-s))d2 _ b(Ca-(ua(Pq,Pb)-r)) _(Pa-qa)cl 2r 2s 2r 2r det+ 2(P-q)clc2 + (Pb-qb)dld2 d 2r -[ 2s 6c2(Pa-qa) (Cb-(Ub(Pa,Pb)-))dl _ (Pa-qa)c (Pb-qb)d2 2r s 2r 2s det He > 0 As discussed above, the denominator will always be positive. When we calculate the determinant in the numerator and simplify, we obtain 6(Cb-(ub(PaPb)-s)(Pa-qa)(cd2-2d2) + (Pa-qa)(Pb-qb)(cld -C2dld2) 2rs 4rs 19

+ 6(Ca-(ua(Pa,Pb)-r)(Cb-(Ub(Pa,Pb)-s)dl + (Ca-(ua(Pa,Pb)-r)(Pb-qb)d > 0 4rs 4rs > From assumption (A5) we know that cl > c2, dl > d2. From from assumption (A6), we know that the capacities for each product will be greater than or equal to the lower bounds on their respective demands. We also know from assumption (A2) that the price of each product must exceed the combined variable production costs and the per unit investment cost for capacity. Consequently, all of the terms in the expression above are positive. Therefore, we conclude that if C increases, the price of product A will increase. The change in the price of product B when C increases is: (Ca-(ua(Pa,Pb)-r))c1 _ (Pa-qa)c (Pb-qb)dg _ 2 R(PaPb) r 2r 2s aPaWC det (Ca-(ua(Pa,Pb)-r))c2 + (Cb —(Ub(Pa,Pb)- s))d2 2r O2s 9R(Pa,Pb) + (Pa-q)ClC2 + (Pb-Qb)dld2 aPbaC 2r - 2 bPb 6P~~~~b = det He > 0 _(Ca-(Ua(Pa,Pb)-))C1 _(Pa-q)c _ (Pb-qb)d2 _ 6(Ca-(ua(Pa,Pb)-r))_ 6(Pa-qa)cl r 2r 2s 2r 2r det (Ca-(ua(PaPb)-r))c2 + (Cb-(Ub(Pa,Pb)-s))d2 (Pa-a)clC2 (Pb-qb)dld2 2r + 2r + 2s det He > 0 For the numerator in this case we obtain the following expression: (Pa-qa)(Pb-qb)(cidld2-c2d2) 6(Ca-(Ua(Pa,Pb)-r))2c2 (Pa-qa)(Cb-(Ub(Pa,Pb)-))Cld2 4rs + 42 2rs + (Ca-(ua(Pa,Pb)-r))(Cb-(ub(Pa,Pb)-s))d2 + 5(Pb-qb)(Ca-(ua(Pa,Pb)-r))Cld2 > 0 4rs 4rs Since cl > c2, d1 > d2 by assumption (A5) and the prices of each product are greater than the variable production costs as discussed in the proof of bpa above, the first term of the expression will always be positive. All of the other terms will also be positive since the capacities of both products will be greater than or equal to their respective lower bounds by assumption (A6). Therefore, we conclude that if C increases, the price of product B will also increase. E Example 3: If Ca increases, then any of the following cases are possible: 20

Case (i): Pa and Pb decrease In this example let: Ca = 1000, Cb = 1000, qa = 2, qb = 2, C= 2000, c1 = 50, c2 = 35, r= 400, D= 3000, di = 50, d2 = 35 and s= 500. In this case the optimal solution is Pa = 98.03, Pb = 109.28 and R(Pa, Pb) = 174435.5. If we increase Ca from 1000 to 1001, we observe that with the set of parameters above, Pa decreases to 98,Pb decreases to 109.27 and R(Pa, Pb) increases to 174474.30. Case (ii): Pa and Pb increase In this case suppose: Ca = 1000, Cb = 1000, qa = 2, qb = 2, C= 2000, ci = 1000, C2 = 18, r= 400, D= 3000, di = 21, d2 = 19 and s= 1000. In this case the optimal solution is Pa = 2.24, Pb = 90.91and R(Pa, Pb) = 72452.92. In this case, increasing Ca to 1001, we find that the optimal Pa increases to 3.14, the optimal Pb increases to 92.22 and R(Pa, Pb) increases to 73461.97. Case (iii): Pa decreases and Pb increases For this example let: Ca = 1000, Cb = 1000, qa = 2, qb = 2, C= 2000, ci = 60, c2 = 50, r= 400, D= 3000, di = 1000, d2 = 19 and s= 250. In this case the optimal solution is Pa = 21.71, Pb = 2.86 and R(Pa, Pb) = 16293.95. By increasing Ca to 1001, we see that Pa decreases to 21.70, Pb increases to 2.860694 and R(Pa, Pb) increases to 16299.87. Note that the case where Pa increases and Pb decreases is not possible because this strategy would adversely affect the number of product A being sold and thus would not help the firm take advantage of the additional capacity. In fact, if the firm were able to improve its profits using this strategy, it would have done so before adding more capacity. Since our problem is symmetric in A and B, we see that Cb can also exhibit the same behavior. E Example 4: When the production costs of the products change, we have more than one potential outcome. We show below examples how changes in qa affect our optimal pricing strategy. Given the symmetry of the problem, the same outcomes can occur for qb. 21

As the product cost of product A (qa) increases, we have the following possible cases: Case (i): Pa and Pb increase In this example let: Ca = 1000, Cb = 1000, qa = 2, qb = 2, C= 2000, cl = 50, c2 = 35, r= 400, D= 3000, di = 50, d2 = 35 and s= 500. In this case the optimal solution is Pa = 98.03, Pb = 109.28 and R(Pa, Pb) = 174435.5. If we increase qa from 2 to 3, we observe that Pa increases to 98.15, Pb increases to 109.35 and R(Pa, Pb) decreases to 173578.5. Case (ii): Pa increases and Pb decreases For this example let: Ca = 1000, Cb = 500, qa = 2, qb = 1, C= 2000, ci = 50, c2 = 49, r= 400, D= 1000, di = 51, d2 = 49 and s= 250. In this case the optimal solution is Pa = 588.41, Pb = 576.96 and R(Pa, Pb) = 696029.8. In this case, increasing qa to 3, we find that the optimal Pa increases to 588.42, the optimal Pb decreases to 576.95 and R(Pa, Pb) decreases to 695218.8. It is clear that the case where both prices decrease is not possible since it would only adversely affect the profitability of both products. The case where Pa decreases and Pb increases is also not possible since this strategy would have been pursued before the increase in production costs if it were possible to increase the profit function's value. E From proposition 6, we discover that changes in the price elasticities of the products have different effects on the optimal pricing strategy. When a product's demand becomes more sensitive to changes in its price, the optimal strategy is to decrease the price of both products. On the other hand, if the cross price elasticity increases, the optimal prices of both products increases. When production costs change, the optimal pricing scheme will be different depending on the parameters of the mean demand functions and the available capacities. We observe that when production costs increase a firm will either increase the price of the product whose cost has increased and decrease the price of the other or increase both products' prices. Similarly, the parameters of the model affect the optimal pricing strategy when the capacity of one the products is changed. In the case of an increase in capacity, we have shown cases where a firm may increase or decrease the price of both of 22

its products or it may decrease the price on the product that has the additional capacity while increasing the price of the other. 5 Individual-Product Optimal Decisions In the previous sections, we examined cases where a firm makes decisions about both of its products' capacities/prices simultaneously. We also assumed that the firm was trying to maximize overall profits (from both product lines). However, it is often the case that firms have separate brand managers for each product line and that these brand managers are evaluated based upon the profitability of their product line alone. In that case, the brand manager will make decisions in order to maximize profits of his or her own product line alone rather than to maximize overall profits for the firm. In this section, we analyze how each of the decisions analyzed in Sections 2-4 would be changed by the fact that they are made by managers trying to maximize individual-product optimal decisions rather than globally optimal decisions maximizing the sum of the profits from the two product lines. We therefore differentiate between the globally optimal decisions in Sections 2-4 and the individually optimal decisions in this section. We also note that in this section we pay attention to the order in which decisions are made and announced. We will show below that this order is significant when both managers can only make pricing decisions but the order does not affect the eventual decisions otherwise. We explore various assumptions about the managers' behavior as in the classical Cournot, Stackelberg, and collusion models of duopoly. 5.1 Capacity decisions We return to the case in Section 2 where the firm determines the optimal capacities for both of its products; however, we now assume that products A and B have their own brand managers. Regardless of whether manager A or B makes its capacity decision first; the profit function for the manager of product A is: 23

Ca Rl(Ca) = (P- qa) f Xa fa(Pa, Pb, xa)dxa Ua(Pa,Pb)-r ua(Pa,Pb)+r +(Pa - qa) Ca f f fa(Pa, Pb, za)dxa - ia. Ca where fa(Pa, Pb, Xa) = 2 and Xa Ca is the demand for product A. Similarly, product B's profit function is defined as: Cb ua(Pa,Pb)+S R2(Cb) = (Pb-qb)' f Xb'fb(Pa, Pb, Xb)dXb +(Pb-qb)'Cb' f fb(Pa, Pb, Xb)dXbUb(Pa,Pb)-S Cb ib' Cb fb(Pa, Pb, Xb) = 2 and Xb is the demand for product B. Proposition 7 The individually optimal Ca and Cb are the same as the globally optimal Ca and Cb of Proposition 1. Proof: The result directly follows from the fact that the first order conditions for Ca and Cb are the same as those in Section 2. 0 Proposition 7 states that if brand managers can only make capacity decisions, even if they make individually optimal decisions, they will end up maximizing global profits as one manager's capacity decision does not affect the other's decision. In the next subsections, however, we show that when pricing decisions are involved (which affect the other product's demand as well), individually optimal decisions differ from globally optimal decisions. 5.2 Deciding on the price of A and the capacity level of B We now examine the case where the optimal price of A, P*, and the optimal capacity for B, C*, is determined. The manager maximizing product A's profits has the following objective function: Ca R (Pa) = (Pa - qa) f a fa(Pa, Pb, xa)dxa Ua(Pa,Pb)-r ua(Pa,Pb)+r +(Pa - qa) ' Ca f fa(Pa, Pb, Xa)dXa, Ca where: fa(Pa, Pb, Xa) = r and Xa is the demand for product A. Note that the manager of product A does not care about the decision that the manager of product B makes about the capacity for product B. However, the manager of product B 24

does care about the price of product A as this affects the demand for product B. Therefore, the optimal Cb is a function of the optimal Pa which is obtained by solving the first order condition above. Product B's profit function is defined the same as R2(Cb) in the previous subsection. The first order condition of R1(Pa) is: Rl(P) _ [C2-(ua(Pa,Pb)-r)21 + (Pa-qa)-[Ca-(ua(Pa,Pb)-r)]. Ua(PaPb) aPa - 2r + 2r Pa + Ca-[ua(PaPb)+rCa] = 0. Rearranging terms and substituting in the functions for ua(Pa, Pb) and ub(Pa, Pb) we can obtain the optimal value of Pa. Comparing the optimal Pa in this case with the optimal Pa derived in Section 3, we clearly see that the optimal pricing strategy for product A differs depending on whether individually optimal or globally optimal decisions are being made. The following example shows globally optimal price and capacity levels can be significantly different than individually optimal levels. Example 5: Suppose that Ca = 1700, qa = 3, qb = 2, C= 2000, cl = 60, c2 = 50, r= 400, D= 3000, dl = 55, d2 = 40, s= 250, ib = 1 and Pb = 77.98. We obtain a globally optimal solution of Pa = 76.38, Cb = 2009.886 and R(Pa, Cb) = 228761.18. Product A's profits in this case are 94560.83. For the same parameters, the individually optimal solutions are: Pa = 68.82 with a profit for product A of R(Pa) = 105701.13 and consequently Cb = 1707.557 with R(Cb) = 111224.07 resulting in a total profit of 216925.20. We see in this case that the manager of product A can improve the profitability of product A alone by decreasing its price compared to the globally optimal price of product A; however this decreases the sum of the profits from both products. 5.3 Pricing decisions In Section 4, we discussed the case where a firm simultaneously sets the prices for both of its products. We now examine the case where managers for products A and B set prices in order to optimize individual product profits. Let us assume that the price of product A 25

is set first. The profit function for product A will be the same as the previous subsection R1(Pa). However, as we can see, the optimal price for A depends on what price will be chosen for product B. Since we assume that all pricing and capacity information is known, the decision maker for the price of product A can predict what product B's optimal pricing strategy will be after the decision is made regarding the price of A. Product B's profit function is defined as: Cb R2(Pb) = (Pb - qb) f Xb - fb(Pa, Pb, Xb)dXb Ub(Pa,Pb)-s Ua(Pa,Pb)+S +(Pb - qb) ' Cb f fb(Pa, Pb, Xb)dxb - ib * Cb, Cb where fb(Pa, Pb, xb) = 2 and xb is the demand for product B. Therefore, before determining product A's optimal price, we solve product B's problem. The first order condition of R2(Pb) is: OR2(Pb) _[C,2-(b(Pa,Pb)-s)2] (Pb-qb).[Cb-(Ub(Pa,Pb)-s)] Ob(PaPb) aPb - - 2s + 2s ' OPb + Cb-[Ub(Pa,Pb)+S-Cb] = 0 2s From the equation above, we can obtain an expression for the optimal price for product B, which may then be used to find the optimal price for product A. To find the optimal price for product A, we have three different approaches available: (i) the Cournot model, (ii) the Stackelberg model, and (iii) the collusion model. In the Cournot model, each product manager assumes that the price of her or his product does not affect the price of the other. In the Stackelberg, one manager follows the other's lead in setting price. In the collusion model, both managers act together to establish best prices for overall profit. (i) Cournot model In this approach, the manager of product A assumes an optimal price of product B, P*, as a parameter in the first order condition of R1(Pa) to obtain: Ri(Pa) _ [C~-(ua(Pa,P,)-r)2] (Pa-qa) [Ca-(Ua(PaP )-r)] aUa(PaP) PPa - 2r + 2r aPa Ca-[Ua(Pa,P )+r-Ca] 0O 2r The optimal solution, P*, was shown in Subsection 5.2 where Pb is now substituted with Pb. The manager of product B follows the same procedure to obtain a price with Pa as a 26

parameter. Solving the simultaneous equations for the two prices produces the result. (ii) Stackelberg model For this approach, the manager of product A assumes that the manager of product B is the price setter. We substitute the optimal P* as a function of Pa directly into the function Rl(Pa) to obtain: Ca R (Pa) = (Pa - qa) f Xa ' fa(Pa, P^, Xa)dXa Ua(Pa,Pb)-r ua(Pa,Pb)+r +(Pa - qa). Ca f fa(Pa, P, a)dxa. Ca The first order condition for this function clearly differs from the Cournot approach shown above since P* is dependent on Pa. Therefore, the optimal price for A differs from the price obtained in approach (i). We now illustrate the above approaches. For analytical simplicity, we examine the case where both products have unlimited capacities and the mean demand functions for products A and B are ua(Pa, Pb) = C - c1Pa + c2Pb and ub(Pa, Pb) = D - dlPb + d2Pa respectively and where cl > c2, dl > d2, cl > d2, and dl > c2. (i) Cournot model We first begin by finding the optimal price of product B. In this case, product B's profit function is defined as: R2(Pb) = (Pb - qb) ' Ub(Pa, Pb). The first order condition is: ub(Pa, Pb) + (Pb-q (Pa, Pb) = 0. oPb Rearranging terms, we obtain: D + d2Pa + qbdl Pb~- = 2dl We now solve for the optimal price for product A. The profit function for product A is: RI1(Pa) = (Pa - qa) Ua (Pa, Pb) 27

The first order condition is then: ua(Pa, P ) + (Pa - a) a(Pa Pb). oPa Substituting Pb into the equation above and solving for Pa we obtain the optimal price of A to be: Dc2 + 2Cdi + 2cldlqa + c2dlQb a 4cldl - c2d2 (ii) Stackelberg model We again begin by finding the optimal price of product B. The profit function for product B is the same as in the Cournot model above and hence the optimal price of B, P*, will be same expression. The profit function for product A is now defined as: R (Pa) = (Pa - qa) ' ua(Pa, Pb). Substituting Pb* into the profit function and solving for Pa in the first order condition, we now have the optimal price of A to be: Dc2 + 2Cdl + 2cldlqa + c2dlqb - c2d2qa a 4cldl - 2c2d2 (iii) Setting both prices simultaneously(collusion) We now return to the model presented in Section 4, where the product managers agree to set both prices simultaneously. In other words, there is collusion in setting the prices for both products in order to optimize system wide profits. We present below the optimal prices for the products where both products have no capacity constraints. The profit function in this case is: R(Pa, Pb) = (Pa - qa) * Ua(Pa, Pb) + (Pb - qb) ' Ub(Pa, Pb). The first order conditions are: 28

,ua(Pa,.Pb) = 0 ua(Pa, Pb) + (Pa - qa) Oua(Pa, Pb) o oPa Ub(Pa, Pb) + (Pb - qb) b(Pa Pb) = 0. aPb Solving the equations for Pa and Pb we obtain: * C + clqa - d2qb + (c2 + d2) (2c(D - 2qa + dlqb) + (C2 + d2)(C + clqa -d2qb)) a- 2cl 2c (4cldl - (c2 + d2)2) and p (c2 + d2) ' (2cl(D - C2qa + dlqb) + (c2 + d2)(C + clqa - d2qb)) 6~~~b -4cldl -(c2 + d2)2 From the expressions in (i), (ii), and (iii) we observe that the optimal prices obtained from the model depend on the price elasticity and cost parameters. We provide below examples where the optimal prices are different relative to each other based on these parameters. Example 6: Collusion prices > Stackelberg prices > Cournot prices. Suppose qa = 2, qb = 2, C= 2000, c1 = 60, c2 = 30, D= 2000, d1 = 60 and d2 = 20. In this case we obtain the following results: Collusion Stackelberg Cournot Pa = 29.63 24 23.04 Pb = 29.51 21.67 21.51 Example 7: Stackelberg prices > Cournot prices > Collusion prices. In this case let qa = 2, qb = 200, C= 2500, c1 = 175, c2 = 2, D= 2000, d1 = 79 and d2 = 10. The resulting prices are: Collusion Stackelberg Cournot Pa= 8.52 11.109 11.107 Pb= 117.08 117.1296 117.1295 29

Example 8: (Collusion price of A > Cournot price of A > Stackelberg price of A) and (Cournot price of B > Stackelberg price of B > Collusion price of B). For this example let qa = 200, qb = 2, C= 2500, ci = 175, c2 = 2, D= 2000, d1 = 79 and d2 = 10. The optimal prices are: Collusion Stackelberg Cournot Pa = 107.75 107.23 107.26 Pb= 19.31 20.444 20.447 From these examples, we may conclude that the timing of pricing decisions, the optimization of individual versus system wide profits along with the relative sizes of the price elasticities, and costs can have different impacts on a firm's optimal pricing strategy. 6 Conclusions and Further Research In this paper, we addressed joint capacity and price decisions for substituable products. We have shown that pricing and capacity decisions are highly affected by the actual parameters that the decision makers can control. The presence of decision makers who optimize individual channel profits versus optimizing system wide has a significant impact on the decisons made. Many research questions on pricing and capacity setting for substitutable products remain open. For example, the case where a firm has control over both products' prices and capacities remains open. (Preliminary research shows this problem to be extremely challanging.) Furthermore, we have only considered a single period problem in this paper. Further research should consider multiple period problems. Bibliography [1] Bassok, Y., R. Anupindi, and R. Akella, "Multiperiod Inventory Models with Substitution," 30

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