Buying an Input from a Competitor with a Production Learning Curve Matthew Selove∗

Dominique Lauga†

January 31, 2018

Abstract Many firms buy a production input from a competitor, but managers often worry that this type of supply agreement could help the competitor develop a strategic advantage. We develop a model in which two firms compete during two periods, and in each period, the focal firm can buy an input either from its competitor or from a third party. Buying from the competitor in the first period helps the competitor move down its learning curve and produce the input at lower cost in the second period. Such a cost reduction implies the competitor can credibly threaten to set a lower goods market price if the focal firm buys its input from the third party, which strengthens the competitor’s bargaining position and can reduce the focal firm’s equilibrium profits. Despite concerns about weakening its future bargaining position, the focal firm’s desire to avoid intense price competition in the first period can cause it to make an even larger fixed payment for the input than it would in the absence of a possible production breakthrough.

∗ †

University of Florida, [email protected] University of Cambridge, [email protected]

1

Introduction

It is common for a firm to buy a production input from its competitor.

Such

supply arrangements typically arise because the competitor can produce an input more efficiently than other suppliers could. For example, Apple and Samsung are the two largest competitors in the smartphone market, and yet Apple’s iPhone uses microchips that it buys from Samsung, which has particular expertise at creating this type of chip. However, managers often express concern that buying an input from a competitor could help the competitor develop a strategic advantage. In fact, Apple has started looking for a new supplier of smartphone chips, and according to the Wall Street Journal, “Apple executives have expressed concern that their dependence on Samsung limits Apple’s ability to control its destiny by constricting Apple’s negotiating power and ability to use different technologies” (Lessin et al. 2013). This paper develops a game theoretic model to explore how production learning curves affect a firm’s incentives to buy an input from its competitor. In our model, two competing firms engage in Nash bargaining over the variable and fixed payments that one firm makes to the other for a production input. If they reach an agreement, the focal firm buys its input from the competitor, but if they fail to reach an agreement, the focal firm buys its input from a third party. The focal firm and its competitor then engage in price competition in the goods market. This entire process is then repeated in a second production period, before which the competitor has some probability of making a production breakthrough that allows it to produce the input at a reduced marginal cost. Buying from the competitor helps soften price competition. Because a competitor that supplies an input earns a profit from each unit of the good that the focal firm sells, it has less incentive to undercut the focal firm’s price in the goods market. 2

Buying from the competitor in the first period also allows the competitor to produce a higher volume of the input, which increases the probability that it makes a production breakthrough that reduces its marginal cost of producing the input in the second period. Such a marginal cost reduction increases industry profits, but also weakens the focal firm’s bargaining position. As the competitor becomes more efficient at making inputs, there is a shift in the relative attractiveness of each firm’s outside option. If the two firms fail to reach an agreement on the input price, in which case the focal firm is forced to buy its inputs from a third party, then the competitor’s ability to produce its own inputs at lower marginal cost provides it with an advantage in the goods market competition. This threat makes the focal firm eager to reach a supply agreement with its competitor in the second period, thus allowing the competitor to extract a larger fixed payment for the input, which can make the focal firm worse off than it would be if the competitor had higher production costs for the input. In the first period, firms must consider how their decisions affect the probability of a production breakthrough, and thus their expected second-period profits. If the focal firm buys from the third party in the first period, firms engage in intense price competition in the goods market, as the competitor seeks to increase its probability of a production breakthrough and the focal firm tries to prevent such a breakthrough. Surprisingly, the focal firm’s desire to avoid this intensified price competition implies it may make an even larger fixed payment for the input than it would in the absence of a possible production breakthrough, even though buying from the competitor increases the probability of such a cost reduction. Thus, our model provides one possible explanation for why a firm may buy a production input from its competitor even as its managers worry about the long run strategic advantage this supply arrangement provides to the competitor.

3

Section 2 discusses related literature.

Section 3 presents the formal model.

Section 4 presents conclusions. The appendix contains proofs of all results.

2

Related literature

Previous literature has developed game theoretic models in which a firm can supply an input to a competitor (Venkatesh et al. 2006; Xu et al. 2010; Pun 2014). Consistent with our findings, these papers have shown that supplying an input to a competitor helps avoid intense price competition in the goods market. A key difference in our paper is that we incorporate learning curve effects and multiple periods into our model to show how buying from a competitor in one period affects future negotiations over the input price. Other papers study related problems, but with a different focus and different modeling assumptions. Lim and Tan (2010) develop a model in which a monopolist decides the proportion of demand to outsource in period one, and its supplier can then enter the goods market as a competitor in period two. In our model, both firms compete in the goods market in both periods, which implies, for example, that sourcing from the competitor in period one softens price competition in that period, and that competing firms’ price decisions in period one affect the probability of a production breakthrough. Besanko et al. (2014) also develop a model in which a firm’s sales volume affects the probability of a production breakthrough, which can cause an incumbent to set lower prices to deter entry. In our model, the competitor supplies an input to the focal firm, which implies it can make a production breakthrough not only based on sales of its own products, but also based on sales of inputs it supplies for the focal firm. More generally, our paper is related to the large marketing literature that uses game theoretic models to study channel coordination (e.g., Jeuland and Shugan 1983; 4

Iyer 1998; Iyer and Villas-Boas 2003; Desai et al. 2004; Dukes et al. 2006; Amaldoss and Shin 2015). Because one firm in our model is both a supplier and a competitor, the classic double marginalization problem studied in this literature actually benefits the firms in our model by softening price competition.

3

Model

Assume two firms can each sell a product to a unit mass of customers who are uniformly distributed along a line of length one. The firms are fixed at opposite sides of the line, with firms indexed by i ∈ {s, b}. Each customer has product valuation V , faces positive transportation cost t per unit of distance, and will buy at most one unit of the product in each period j ∈ {1, 2}. Pi,j denotes Firm i’s price for the product in period j, and Di,j denotes the Firm i’s demand in period j. In each period j, a customer at location φ derives utility V − tφ − Ps,j from buying Firm s’s product, and utility V − t(1 − φ) − Pb,j from buying Firm b’s product. Customers make purchase decisions to maximize their utility in each period. Each unit of the product requires a production input, and Firms s and b are a potential supplier and buyer of this input, respectively. In period j, Firm s can produce the input at marginal cost Cj . Firm s can use these inputs in its own products, and it can also supply inputs to Firm b for a per-unit price Wj plus a fixed payment Fj , where Wj ≥ 0, and Fj can be either positive or negative. Note these input contracts consists of a standard two-part tariff. Firm b does not have the capability to create the input on its own, but it can also purchase an identical input from a third party at an exogenous per-unit price c , where W c > Cj . In principle, we could explicitly model the pricing decision of W the third party. The price assumption used here would be consistent with multiple

5

outside parties engaged in perfect competition to sell the input. Aside from the cost of acquiring or producing the input, other marginal production costs are normalized to zero. Given this set-up, if Firm b buys the input from the third party in period j, firms’ profit functions πs,j and πb,j are given by:

πs,j = (Ps,j − Cj )Ds,j

(1)

c )Db,j πb,j = (Pb,j − W

(2)

On the other hand, if Firm b buys the input from Firm s, firms’ profit functions are:

πs,j = (Ps,j − Cj )Ds,j + (Wj − Cj )Db,j + Fj

(3)

πb,j = (Pb,j − Wj )Db,j − Fj

(4)

In the first period, Firm s has marginal cost CH for the input. In the second period, its cost is reduced to CL with probability α(N1 ) and remains at CH with probability 1 − α(N1 ), where N1 is the total units of the input Firm s produces in period one, and 0 < CL < CH . The function α is increasing in N1 , which implies the probability of a cost reduction increases as Firm s produces more units of the input in period one. For analytical tractability, we assume α is a linear function given by:

α(N1 ) = α0 + α1 N1

(5)

where α0 ∈ [0, 1), α1 > 0, and α0 + α1 ≤ 1. Using this linear probability function allows us to compute closed form results for equilibrium prices and profits in the first period, when firms must account for how their decisions affect the probability of a

6

production breakthrough by Firm s. Firms engage in Nash bargaining over the input prices at the start of each period, choosing variable price Wj to maximize industry profits and fixed price Fj to determine how they split those profits. If they reach an agreement, Firm b will buy its inputs from Firm s in the current period, and if they fail to reach an agreement, Firm b will buy its inputs from the third party in the current period. Each firm seeks to maximize its total profits over the two periods. Table 1. Overview of notation V

Customer valuation for the product

t

Transportation cost per unit of distance

i ∈ {s, b}

Index of competing firms (supplier and buyer of the input, respectively)

j ∈ {1, 2}

Index of time

Pi,j

Price for firm i in period j

Di,j

Demand for firm i in period j

πi,j

Total profits for firm i in period j

c W

Unit price of the input from the third party

Wj

Unit price of the input from the supplier in period j

Fj

Fixed payment the buyer makes to the supplier in period j

Cj ∈ {CH , CL } Nj α(N1 )

Firm s’s marginal cost of producing the input in period j Total units of the input produced by firm s in period j Probability that firm s experiences a cost reduction before period 2

To summarize, the game timing is as follows: 1. Firms engage in Nash bargaining over W1 and F1 . 2. Firms simultaneously set prices Ps,1 and Pb,1 .

7

3. Customers make purchase decisions, and profits are realized, for period one. 4. Firm s experiences a production breakthrough with a probability α(N1 ). 5. Firms engage in Nash bargaining over W2 and F2 . 6. Firms simultaneously set prices Ps,2 and Pb,2 . 7. Customers make purchase decisions, and profits are realized, for period two. Given that consumer prices are typically easier to change that supply contracts, it seems reasonable to assume that, in each period, firms first agree to supply contracts, and consumer prices then reach an equilibrium conditional on those contracts. The game timing given above reflects this assumption. We make the following assumption, which helps ensure that, in equilibrium, the market is covered and both firms have positive demand, regardless of whether Firm b buys its input from Firm s or from the third party: c − 3t ≤ Cj ≤ W c ≤ V − 2t Assumption 1. 0 ≤ W We also make the following assumption, which ensures each firm has positive equilibrium demand in the first period if the Firm b buys from the third party. c − 3t + α(CH − CL ) ≤ CH Assumption 2. W Finally, we make the following assumption, which ensures the market is covered in the first period if the Firm b buys from Firm s.   Assumption 3. α1 (CH − CL ) 3t1 − 12 <

3.1

t 2

Results for Period Two

We solve the model by backward induction. This section derives results for period two conditional on the Firm s input production cost, C2 . The next section then solves 8

for results in period one, when firms must account for how their decisions affect the probability of a reduction in this period two input cost. We first analyze the subgame that results if firms fail to reach an agreement on the input contract for period two, in which case Firm b buys its inputs from the third party. Lemma 1. If Firm b buys its input from the third party in period two, equilibrium prices and profits are: 2 Ps,2 = t + C2 + 3

1c W 3

(6)

1 Pb,2 = t + C2 + 3

2c W 3

(7)

πs,2 =



2 t 1 c 1 c W − C2 + W − C2 + 2 3t 18t

(8)

πb,2 =



2 1 c t 1 c − W − C2 + W − C2 2 3t 18t

(9)

c > C2 , which implies Firm s has lower costs and higher equilibrium Recall that W profits than Firm b in this subgame. As the Firm s input cost C2 decreases, the difference in equilibrium profits between the firms grows, which provides Firm s with a stronger bargaining position as firms negotiate over the input price. We next analyze the subgame that results if firms agree on an input contract, for an variable input price that we will show is optimal in the sense of maximizing industry profits. Lemma 2. If Firm b buys its input from Firm s with variable input price W2 = V − 23 t in period two, equilibrium prices and profits are:

Ps,2 = V − 9

t 2

(10)

Pb,2 = V −

t 2

(11)

πs,2 = V − t − C2 + F2

πb,2 =

(12)

t − F2 2

(13)

For the variable input fee specified in this lemma, both firms set goods market price equal to customers’ product valuation minus the transportation cost from either firm to the middle of the line. These prices imply firms divide the market evenly, and the marginal customer in the middle of the line is indifferent between purchasing and not purchasing, which is the outcome maximizes total profits for the two firms. Firm b has no incentive to undercut this price because the marginal input price, W2 , is set high enough to make such a price reduction unprofitable, and Firm s has no incentive to undercut this price because doing so would reduce its profit from sales of the input to Firm b. Thus, this supply arrangement allows firms to coordinate goods prices to maximize industry profits. Under Nash bargaining, firms choose a fixed price for the input such that each firm’s equilibrium profits equal its profits from the case in which the fail to reach an agreement plus one half of the increase in industry profits that results if they do reach an agreement. This outcome is formally stated in the following proposition. Proposition 1. In the equilibrium of period two, firms agree on an input contract with W2 = V − 32 t, which results in both firms setting goods price V −

10

t 2

and total

industry profits of V − 2t − C2 . Under Nash bargaining, the fixed fee is: F2 =


1 c 3t C2 V + − + W − C2 4 2 2 3t

(14)

The resulting equilibrium profits are as follows:
V − 2t − C2 1 c + W − C2 2 3t

(15)


V − 2t − C2 1 c − = W − C2 2 3t

(16)

πs,2 =

πb,2

Note that a decrease in the input cost C2 unambiguously increases equilibrium profits for Firm s. However, if t < 32 , then a decrease in C2 decreases equilibrium profits of Firm b. Intuitively, if the degree of differentiation between firms is relatively low (t is small), then low input production costs for the seller imply the seller’s profits are much higher than the buyer’s profits if the latter buys its inputs from the third party. Despite the increase in total industry profits from an input cost reduction, the resulting shift in firms’ bargaining positions leaves the buyer worse off than it would be if the seller had higher input costs.

3.2

Results for Period One

We now derive the equilibrium for period one, when firms must consider how their decisions affect both first-period profits and expected second-period profits. Recall that the probability of a production breakthrough before period two is α0 + α1 N1 , where N1 is the number of units of the input Firm s produces in period one. If such a breakthrough occurs, Firm s’s period 2 input cost is reduced from CH to CL . Based on the derivations in the previous section, expected period-two profits conditional on N1 are the following: 11

  1 1 E[πs,2 |N1 ] = πs,2 (CH ) + (α0 + α1 N1 )(CH − CL ) + 2 3t

(17)

  1 1 E[πb,2 |N1 ] = πb,2 (CH ) + (α0 + α1 N1 )(CH − CL ) − 2 3t

(18)

where πi,2 (CH ) denotes Firm i’s equilibrium period-two profits given input cost CH . We now introduce additional notation to simplify the exposition of our results. Let ∆i denote the marginal effect of an increase in N1 on Firm i’s expected second-period profits:   1 1 ∆s = α1 (CH − CL ) + 2 3t 

1 1 ∆b = α1 (CH − CL ) − 2 3t

(19)

 (20)

Note that ∆s > 0, which implies that an increase in N1 increases expected period-two profits for Firm s. On the other hand, if if t <

2 3

then ∆b < 0, in which case an

increase in N1 decreases expected period-two profits for Firm b. For each firm, maximizing expected profits is equivalent to maximizing the following functions, which we will refer to as the period-one objective functions.

π es,1 = πs,1 + ∆s N1

(21)

π eb,1 = πb,1 + ∆b N1

(22)

These objective functions reflect each firm’s period one profits plus the expected impact of period-one decisions on the firm’s period-two profits. We first derive the equilibrium of the subgame in which Firm b buys from the third party in period one.

12

Lemma 3. If Firm b buys its input from the third party in period one, equilibrium prices and objective functions are:

π es,1 =

π eb,1 =

2 1 c Ps,1 = t + (CH − ∆s ) + (W + ∆b ) 3 3

(23)

1 2 c Pb,1 = t + (CH − ∆s ) + (W + ∆b ) 3 3

(24)



2 t 1 c 1 c + W − CH + ∆ s + ∆ b + W − CH + ∆ s + ∆ b 2 3 18t



2 1 c t 1 c − W − CH + ∆ s + ∆ b + W − CH + ∆ s + ∆ b + ∆ b 2 3 18t

(25)

(26)

Note that an increase in ∆s and a decrease in ∆b imply firms engage in more intense price competition. Because ∆s > 0, Firm s tries to increase the probability of a production breakthrough by reducing its price so it produces more inputs, and if ∆b < 0 Firm b tries to prevent such a breakthrough by reducing its price to reduce the number of inputs Firm s produces. Also, because ∆s + ∆b > 0, which implies ∆s > −∆b , in this equilibrium, concerns over a potential production breakthrough lead to a larger reduction in the equilibrium price of Firm s than in the equilibrium price of Firm b. Thus, the equilibrium market share of Firm s is greater than it would be if firms simply maximized period-one profits. We next analyze the subgame that results if firms agree on an input contract in period one, for the same variable input price that was considered in Lemma 2. Lemma 4. If Firm b buys its input from Firm s with variable input price W1 = V − 32 t in period one, equilibrium prices and objective functions are:

Ps,1 = V − 13

t 2

(27)

Pb,1 = V −

t 2

π es,1 = V − t − CH + F1 + ∆s t − F1 + ∆ b 2

π eb,1 =

(28) (29) (30)

Note these equilibrium prices are the same as in Lemma 2. When Firm b buys from Firm s, as long as the market is covered, Firm s always produces one unit of the input regardless of the prices firms set. Thus, in this case, the possibility of a production breakthrough does not affect equilibrium prices, and the price equilibrium is the same as in the period-two case. We show that this outcome, in which Firm b buys inputs from Firm s, and both firms set goods market price V − 2t , maximizes the sum of firms’ period-one objective functions. These prices maximize period-one profits, and because ∆s +∆b > 0, Firm b producing one full unit of the input also maximizes expected industry profits in the second period. Proposition 2. In the equilibrium of period one, firms agree on an input contract with W1 = V − 23 t, which results in both firms setting goods price V − 2t . In equilibrium, the sum of firms’ first-period objective functions is:

π es,1 + π eb,1 = V −

t − CH + α1 (CH − CL ) 2

(31)

Under Nash bargaining, the fixed fee is:  
3t V CH 1 c 1 1 F1 = − + + W − CH + α1 (CH − CL ) − 4 2 2 3t 6t 4

14

(32)

The resulting equilibrium objective functions are as follows:

π es,1

 
V − 2t − CH 1 c 1 1 = + W − CH + α1 (CH − CL ) + 2 3t 4 2t

(33)

π eb,1

 
V − 2t − CH 1 c 1 3 = − W − CH + α1 (CH − CL ) − 2 3t 4 2t

(34)

In order to show how model primitives drive the equilibrium outcome, we have stated this result directly in terms of the underlying model parameters, and not in terms of ∆s and ∆b . Recall that, if t < 23 , then a production breakthrough by Firm s results in lower equilibrium profits for Firm b in the second period. As we would expect, under this condition, Firm b has a lower equilibrium objective function than it would in the absence of a possible production breakthrough, as reflect by the last term in the equation for π eb,1 being negative. Somewhat surprisingly, if t < 32 , then Firm b makes a larger fixed payment for the input than it would in the absence of a potential production breakthrough, as reflected by the last term in the expression for F1 being positive. This result occurs because Firm b is particularly eager to avoid buying from the third party, knowing that doing so would result in intense price competition, a relatively large market share for Firm s, and relatively low equilibrium profits for Firm b. Thus, Firm b faces a strong incentive to buy from its competitor in the first period even though doing so increases the probability of a production breakthrough that weakens Firm b’s future bargaining position.

15

4

Conclusion

We develop a model in which a firm can buy a production input either from its competitor or from a third party. Buying from the competitor in the first period increases the probability that the competitor moves down its learning curve and has lower production costs in the second period, which leaves the focal firm in a weaker bargaining position. Nonetheless, we show that the focal firm’s desire to avoid intense price competition in the first period, which would result in a relatively lower goods market price and higher market share for the competitor, causes the focal firm to pay a higher first-period price for the input than it would in the absence of a learning curve. Future research could extend our model, for example, with more periods and more possible production cost states, to explore how equilibrium input prices and goods prices change over time as the competitor moves further down its learning curve. Future research could also explore other issues that arise when a firm buys from its competitor, such as the potential for intellectual property leakage as the two firms share information necessary to coordinate production.

16

References Amaldoss, W. and W. Shin (2015). Multitier store brands and channel profits. Journal of Marketing Research 52 (6), 754–767. Besanko, D., U. Doraszelski, and Y. Kryukov (2014). The economics of predation: What drives pricing when there is learning-by-doing? The American Economic Review 104 (3), 868–897. Desai, P., O. Koenigsberg, and D. Purohit (2004). Strategic decentralization and channel coordination. Quantitative Marketing and Economics 2 (1), 5–22. Dukes, A. J., E. Gal-Or, and K. Srinivasan (2006). Channel bargaining with retailer asymmetry. Journal of Marketing Research 43 (1), 84–97. Iyer, G. (1998). Coordinating channels under price and nonprice competition. Marketing Science 17 (4), 338–355. Iyer, G. and J. M. Villas-Boas (2003). A bargaining theory of distribution channels. Journal of Marketing Research 40 (1), pp. 80–100. Jeuland, A. P. and S. M. Shugan (1983). Managing channel profits. Marketing Science 2 (3), pp. 239–272. Lessin, J., L. Luk, and J. Osawa (2013). Apple finds it difficult to divorce Samsung. Wall Street Journal . Lim, W. and S. Tan (2010). Outsourcing suppliers as downstream competitors: Biting the hand that feeds. European Journal of Operational Research 203 (2), 360–369. Pun, H. (2014). Supplier selection of a critical component when the production process can be improved. International Journal of Production Economics 154, 127–135. Venkatesh, R., P. Chintagunta, and V. Mahajan (2006). Research note: Sole entrant, co-optor, or component supplier: Optimal end-product strategies for manufacturers of proprietary component brands. Management Science 52 (4), 613–622. Xu, Y., H. Gurnani, and R. Desiraju (2010). Strategic supply chain structure design for a proprietary component manufacturer. Production and Operations Management 19 (4), 371–389.

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Appendix: Proofs Proof of Lemma 1 If Firm b buys from the third party in period two, firms’ profit functions are:

πs,2

  1 Ps,2 − Pb,2 = (Ps,2 − C2 ) − 2 2t

(35)

πb,2

  1 Ps,2 − Pb,2 c = (Pb,2 − W ) + 2 2t

(36)

Differentiating these functions leads to the following first-order conditions:

Ps,2 =

t Pb,2 + C2 + 2 2

(37)

Pb,2 =

c t Ps,2 + W + 2 2

(38)

Combining these first-order conditions, we have the following equilibrium prices:

Ps,2 = t +

c 2C2 W + 3 3

(39)

Pb,2 = t +

c C2 2W + 3 3

(40)

Assumption 1 ensures each firm has positive demand and the market is covered at these prices. Inserting these prices into the profit functions, we have the following equilibrium profits:

πs,2 =

c − C2 (W c − C2 )2 t W + + 2 3 18t

18

(41)

πb,2 =

c − C2 (W c − C2 )2 t W − + 2 3 18t

(42)

QED Proof of Lemma 2 We will show that Ps,2 = Pb,2 = V − equilibrium given input price W2 = V −

t 2

is the unique price

3t . 2

If firms set prices in a range where the market is covered, their profit functions are: 

πs,2

   1 Ps,2 − Pb,2 1 Ps,2 − Pb,2 = (Ps,2 − C2 ) − + (W2 − C2 ) + + F2 2 2t 2 2t

πb,2

  1 Ps,2 − Pb,2 = (Pb,2 − W2 ) + − F2 2 2t

Taking first derivatives and inserting W2 = V −

3t , 2

(43)

(44)

we have the following first order

conditions: V − 2t − 2Ps,2 + Pb,2 t − 2Ps,2 + Pb,2 + W2 dπs,2 = = =0 dPs,2 2t 2t

(45)

V − 2t − 2Pb,2 + Ps,2 t − 2Pb,2 + Ps,2 + W2 dπb,2 = = =0 dPb,2 2t 2t

(46)

Solving both order conditions, we have Ps,2 = Pb,2 = V − 2t . These first order conditions ensure neither firm would deviate to a lower price in the proposed equilibrium. However, if either firm deviates to a higher price, firms become local monopolists who no longer have the same marginal customer, and the above profit functions no longer hold. Therefore, we need to check that neither firm has an incentive to increase its price. If Firm s deviates to a higher price, setting Ps,2 > V − 2t , its profit function is: 19



πs,2

 V − Ps,2 1 = (Ps,2 − C2 ) + (W2 − C2 ) + F2 t 2

(47)

Taking the first derivative, we have: dπs,2 V − 2Ps,2 + C2 −V + t + C2 = < dPs,2 t t

(48)

where the inequality holds for any Ps,2 > V − 2t . Because Assumption 1 implies C2 < V − t, this derivative is negative, and the firm could increase its profits by lowering its price. If Firm b deviates to a higher price, setting Pb,2 > V − 2t , its profit function is:

πb,2

  V − Pb,2 = (Pb,2 − W2 ) − F2 t

(49)

Taking the first derivative, we have: V − 2Pb,2 + W2 −V + t + W2 dπb,2 = < dPb,2 t t

(50)

where the inequality holds for any Pb,2 > V − 2t . Because a condition of the lemma is W2 = V −

3t , 2

this derivative is negative, and the firm could increase its profits by

lowering its price. Therefore, neither firm has an incentive to deviate from the proposed equilibrium to a higher price.

Finally, these same derivations imply there cannot be any

alternative equilibrium in which either firm sets a price higher than V − 2t , because firms would have an incentive to lower their price until they have the same marginal customer. Inserting the equilibrium prices into firms’ profit functions shows that equilibrium profits are as follows:

20

πs,2

V − 2t − 2C2 + W2 = + F 2 = V − t − C2 + F 2 2

πb,2 =

V − 2t − W2 t − F2 = − F2 2 2

(51)

(52)

QED

Proof of Proposition 1 We first show that the equilibrium derived in Lemma 2, with Ps,2 = Pb,2 = V − 2t , has prices that maximize industry profits. Define total industry profits π2 ≡ πs,2 + πb,2 . If firms set prices in a range where they are local monopolists, industry profits are:     V − Ps,2 V − Pb,2 π2 = (Ps,2 − C2 ) + (Pb,2 − C2 ) t t

(53)

Taking first derivatives, we have: dπ2 V − 2Ps,2 + C2 = dPs,2 t

(54)

dπ2 V − 2Pb,2 + C2 = dPb,2 t

(55)

As shown in the derivation of Lemma 2, these derivatives are negative for prices above V − 2t . Therefore, for any pair of prices in which firms act as local monopolists, industry profits would be increased by one or both firms lowering their prices until they are no longer local monopolists. For prices in which the market is covered, industry profits are:  π2 = Ps,2

   1 Ps,2 − Pb,2 1 Ps,2 − Pb,2 − + Pb,2 + − C2 2 2t 2 2t

21

(56)

Taking first derivatives, we have: dπ2 t + 2Pb,2 − 2Ps,2 = dPs,2 2t

(57)

dπ2 t + 2Ps,2 − 2Pb,2 = dPb,2 2t

(58)

Note that the sum of these derivatives is positive, which implies that as long as the market is still covered, industry profits increase if both firms raise their price by the same amount. Therefore, optimal prices must occur where the marginal customer is indifferent between purchasing and not purchasing, at which point further price increases imply the market is no longer covered. Also note that, if firms set different prices, then the first derivative with respect to the lower price is greater than the derivative with respect to the higher price. Therefore, industry profits increase if the firm with the lower price increases is price and the firm with the higher price decreases its price by the same amount. Together, these observations imply that industry profits are maximized by setting prices equal, with the marginal customer indifferent toward purchasing, which occurs at Ps,2 = Pb,2 = V − 2t . Thus, the equilibrium derived in Lemma 2, based on variable fee W2 = V −

3t , 2

generates prices that maximize industry profits. We will show that, under Nash bargaining, firms agree on the following fixed price, which implies they evenly divide the gains from their supply arrangement:

F2 =


3t C2 V 1 c + − + W − C2 4 2 2 3t

(59)

Inserting this fixed payment into the equilibrium profit functions derived in lemma 2

22

yields the following profits:

πs,2 =


V t C2 1 c − − + W − C2 2 4 2 3t

(60)

πb,2 =


V t C2 1 c − − − W − C2 2 4 2 3t

(61)

Taking these profits minus the profits derived in Lemma 1, we find that each firm’s equilibrium profits given this supply contract exceed the equilibrium profits it would receive if they failed to reach an agreement by the following amount: c − C2 )2 3t C2 (W V − − − 2 4 2 18t

(62)

Note this term is guaranteed to be positive, because we have shown that the equilibrium prices derived in Lemma 2 maximize industry profits, while those derived in Lemma 1 do not. Thus, the proposed variable fee generates equilibrium prices that maximize industry profits, and the proposed fixed fee implies the two firms evenly divide the profit increase from this supply agreement, which is the Nash bargaining outcome. QED

Proof of Lemma 3 If Firm b buys from the third party in period one, firms’ objective functions are: 

π es,1

π eb,1

1 Ps,1 − Pb,1 = (Ps,1 − CH + ∆s ) − 2 2t



    1 Ps,1 − Pb,1 1 Ps,1 − Pb,1 c = (Pb,1 − W ) + + ∆b − 2 2t 2 2t 23

(63)

(64)

Differentiating these functions leads to the following first-order conditions:

Ps,1 =

t Pb,1 + CH − ∆s + 2 2

(65)

Pb,1 =

c + ∆b t Ps,1 + W + 2 2

(66)

Combining these first-order conditions, we have the following equilibrium prices:

Ps,1 = t +

c + ∆b 2(CH − ∆s ) W + 3 3

(67)

Pb,1 = t +

c + ∆b ) CH − ∆s 2(W + 3 3

(68)

Assumptions 1 and 2 ensure each firm has positive demand and the market is covered at these prices. Inserting these prices into the profit functions, we have the following equilibrium profits:

π es,1 =

π eb,1 =



2 t 1 c 1 c + W − CH + ∆ s + ∆ b + W − CH + ∆ s + ∆ b 2 3 18t



2 1 c t 1 c − W − CH + ∆ s + ∆ b + W − CH + ∆ s + ∆ b + ∆ b 2 3 18t

(69)

(70)

QED Proof of Lemma 4 We will show that Ps,1 = Pb,1 = V − equilibrium given input price W1 = V −

3t . 2

24

t 2

is the unique price

If firms set prices in a range where the market is covered, their objective functions are:

π es,1

    1 Ps,1 − Pb,1 1 Ps,1 − Pb,1 = (Ps,1 − CH ) − + (W1 − CH ) + + F1 + ∆s (71) 2 2t 2 2t

π eb,1

  1 Ps,1 − Pb,1 − F1 + ∆b = (Pb,1 − W1 ) + 2 2t

(72)

Note that the terms ∆s and ∆s do not vary based on firms’ period-one prices. When Firm s supplies inputs to Firm b and the market is covered, Firm s always produces one unit of inputs, and marginal price changes to not affect this input quantity. Therefore, the same derivations from the proof of Lemma 2 show that solving both first order conditions implies Ps,1 = Pb,1 = V − 2t . As in the proof of Lemma 2, if either firm deviates to a higher price, firms become local monopolists who no longer have the same marginal customer, and the above profit functions no longer hold. Therefore, we need to check that neither firm has an incentive to increase its price. If Firm s deviates to a higher price, setting Ps,1 > V − 2t , its objective function is: 

π es,1

 V − Ps,1 1 = (Ps,1 − CH + ∆s ) + (W1 − CH + ∆s ) + F1 t 2

(73)

Taking the first derivative, we have: de πs,1 V − 2Ps,1 + CH − ∆s −V + t + CH − ∆s = < dPs,1 t t

(74)

where the inequality holds for any Ps,1 > V − 2t . Because Assumption 1 implies CH < V − t, and the term −∆s is also negative, this derivative is negative, and the firm could increase its profits by lowering its price.

25

If Firm b deviates to a higher price, setting Pb,1 > V − 2t , its profit function is:

π eb,1

  V − Pb,1 1 = (Pb,1 − W1 + ∆b ) + ∆b − F1 t 2

(75)

Taking the first derivative, we have: − t − ∆b V − 2Pb,1 + W1 − ∆b −V + t + W1 − ∆b de πb,1 = < = 2 dPb,1 t t t

(76)

where the inequality holds for any Pb,1 > V − 2t , and the last equality holds because a condition of the lemma is W1 = V −

3t . 2

Assumption 3 ensures −∆b < 2t , which

implies this derivative is negative, and the firm could increase its profits by lowering its price. Therefore, neither firm has an incentive to deviate from the proposed equilibrium to a higher price.

Finally, these same derivations imply there cannot be any

alternative equilibrium in which either firm sets a price higher than V − 2t , because firms would have an incentive to lower their price until they have the same marginal customer. Inserting the equilibrium prices into firms’ profit functions shows that equilibrium profits are as stated in the lemma. QED

Proof of Proposition 2 We first show that the equilibrium derived in Lemma 4 has prices and input production that maximize expected industry profits.

The

same derivations as in the proof of Proposition 1 show that these equilibrium prices maximize total profits in the first period. Furthermore, because ∆s + ∆b > 0, an outcome with input production N1 = 1 maximizes total expected profits in the second period. Thus, the proposed equilibrium maximizes both first-period profits and expected second-period profits. 26

We will show that, under Nash bargaining, firms agree on the following fixed price, which implies they evenly divide the gains from their supply arrangement:

F1 =


3t V CH ∆s 1 c − + − + W − CH + ∆ s + ∆ b 4 2 2 2 3t

(77)

Inserting this fixed payment into the equilibrium profit functions derived in lemma 4 yields the following objective functions:

π es,1 =

π eb,1


V − 2t − CH + ∆s 1 c + W − CH + ∆ s + ∆ b 2 3t


V − 2t − CH + ∆s 1 c = − W − CH + ∆ s + ∆ b + ∆ b 2 3t

(78)

(79)

Taking these objective function values minus the values derived in Lemma 3, we find that each firm’s equilibrium objective function given this supply contract exceed what it would receive if they failed to reach an agreement by the following amount:
2 V 3t CH ∆s 1 c − − + − W − CH + ∆ s + ∆ b 2 4 2 2 18t

(80)

Note this term is guaranteed to be positive, because we have shown that the outcome in Lemma 4 maximize the sum of firms’ objective functions, while those derived in Lemma 3 do not. Inserting the values ∆s = α1 (CH − CL )[ 21 + yields the outcomes stated in the lemma. QED

27

1 ] 3t

and ∆b = α1 (CH − CL )[ 12 −

1 ] 3t