The strategic use of early bird discounts for dealers

Abstract

We study a widely used ordering process (“Early Bird Discounts”) whereby a profit-maximizing manufacturer permits his dealers to place advance orders at a discount before they set retail prices. We show that such discounts may be used to shift just enough channel profits to dealers to enable them to cover their fixed costs and stay in business. If the manufacturer instead simply cut his wholesale price in order to generate gross margins for his dealers, these margins would soon dissipate as price competition among dealers selling the same product forced retail prices back down to the per-unit cost. We show that when dealer fixed costs are low, the manufacturer offers an Early Bird Discount to his multiple dealers that induces all but two of them to exit; when fixed costs are high, the manufacturer offers no preorder discount (i.e. switches to linear pricing) and induces all but one dealer to exit. Although uniform slotting allowances could also be used to reward dealers, a sales-based alternative like an Early Bird Discount sometimes has a key advantage when the manufacturer has dealers in cities of different sizes. If the same Early Bird Discount is offered, dealers in markets with more consumers, who typically have larger fixed costs, will preorder larger amounts and will automatically receive higher gross margins. To duplicate such payments with slotting allowances, non-uniform allowances would have to be offered to firms in different markets, which is divisive and possibly illegal.

Appendices

Appendix A: Colgate-palmolive’s promotion on Ajax laundry detergent

Source: Adapted from Blattberg and Neslin (1990), pp. 319–321.

Note: The terms in this promotion correspond to the notation in our model as follows: θ = 19.90; c = 14.30.

Appendix B: Kaspersky’s 2009 midyear promotion

Following is the 2009 midyear promotion offered by Kaspersky for its sales promotion within the Beijing area of an antivirus consumer software product.

2.1 Beijing region Kaspersky “2010 version” product preorder policy

Respected Dealer Friends,

Kaspersky’s 2010 new full-function security software is soon to be launched. Our preorder policies are as follows:

Pre-order price:

• Kaspersky Full-Function Security Software 2010 Version: 130 yuan/unit

Payment: between July 17–22, 2009^a

Promotional details for Kaspersky Full-Function Security Software 2010 Version

• Each 100-unit order gets 4 additional free units^b

Kaspersky products strictly follow our regional sales policy. Orders from this promotion are restricted to sales within the Beijing region. We will not honor our incentives if your products are found to have been sold in other regions. Please contact our Beijing branch for details of this promotion.

Note: The terms in this promotion correspond to the notation in our model as follows: θ = 130; c = (100/104) (4 free additional units per 100 units as preorder discounts; excluding other discounts from which we abstract).

^a Delivery of the packaged software CDs, based on dealers’ requests, can start as soon as the preordering period ends.

^b In addition, Kaspersky offers other incentives on preorders. These additional terms are described in Kaspersky’s “Dealership-Development” document and may include sales quotas, monetary rewards, prizes, and dealer-specific support in advertising, consumer promotions, and promotional materials.

Appendix C: Computec’s 2005 second quarterly promotions

C1. Computec’s second quarterly promotion in 2005 for its consumer product.

3.1 Computec “new version” product sales policy

Respected Computec Dealers and Distributors:

After several years of hard work and cooperation between Computec and our dealers, Computec’s product has shown growing market shares. Our sales trend is strong and every region is exhibiting hot sales. This makes our product the unassailable leading brand in the market.

To enhance consumer demand in the summer, Computec is launching our “New Version” product, together with millions of dollars of marketing expenses, to organize this promotion campaign. At the same time, to reward our core channel partners and retail outlets during summer holidays, we will start the “1000-store Cool Gift” campaign. We hope our dealers will seize this opportunity to set another sales record.

In this summer promotion campaign, we will stabilize channel prices to protect the profitability of our dealers by penalizing those who viciously lower channel prices.

Our sales policies for the promotion are:

• A. Product and Price

  • Product name: “Computec Consumer Product 2005 Version”

  • List wholesale price: 95 yuan/unit

  • Discounted wholesale price: 90 yuan/unit

• B. Duration

  • July 7–8, 2005 (based on the time shown on your bank telex deposits)

  • From July 9 onward, Computec 2005 Version reverts to its regular list wholesale price of 95 yuan/unit

• C. Ordering Policy

  • Product delivery starts on July 10, 2005 in the order in which we have received payments during the promotion period.^a

  • Computec’s sales managers may assign sales quotas.^b We will offer additional marketing and advertising support for those whose orders exceed these quotas.

Computec Technology Co. Ltd.

Note: The terms in this dealer promotion correspond to the notation in our model as follows: θ = 95; c = 90 (excluding quantity discounts from which we abstracted).

^a Although a dealer has to take all deliveries before a specified date, the dealer can divide his total preorder into smaller portions and decide when to take delivery of each portion.

^b The sales quota includes a minimum order of 1,000 units in this sales cycle.

C2. Computec’s second quarterly promotion in 2005 for its small and medium enterprise (SME) product line

3.2 “Computec product small & medium enterprise (SME) version” product promotion policies

Respected Computec Dealers and Distributors:

Thank you for your support and cooperation for the past many years.

Computec is becoming an integrated manufacturer in the computer accessory market. Our market share and sales are both increasing rapidly for our hardware and software.

The peak selling season has arrived. Again, Computec is organizing our “SME version” promotion. In this promotion, Computec is offering the following incentives for our long-term dealers and distributors:

• 1. Promotion Period

  From 13:30 08/17/2005 to 17:00 08/19/2005 (based on the time shown on your bank telex deposits)

• 2. Ordering price during the promotion period:

  For our assigned dealers, your ordering price is 2% off the standard list wholesale prices.

• 3. Special incentives

  Orders of more than 80000 yuan get 1 unit of Product X^a free.

  • Orders of more than 200000 yuan get 3 units of Product X free.

  • Orders of more than 500000 yuan get 8 units of Product X free.

  • All orders during the promotion period are also eligible for routine annual discounts.^b

• 4 Other regulations

  • Those who ordered from us during this promotion period are eligible to sign an SME agreement with Computec. We will register those company names in “Computec SME dealership list.”^c

  • We may offer advertising and other marketing support for those who participate in this promotion.

• 6. After the promotion period, the ordering prices of “SME version” will revert to the list wholesale prices terms specified in the dealership agreement.

• 7. After we receive your payment, we will start product delivery. Dealers must take possession of all ordered units on or before 17:30 09/02/2005.^d

• 8. Computec reserves all rights to interpret the policies and regulations of this promotion.

  Computec works with you for prosperity.

Computec Technology Co. Ltd.

Note: The terms in this promotion correspond to the notation in our model as follows: θ varies depending on the product; c = 0.98θ (2% preorder discounts; excluding quantity discounts from which we abstracted).

^a Computec also makes and sells an enterprise computer product, referred to here by the pseudonym Product X, which is complementary to the SME product line.

^b Annual discounts are on average 10% and based on preorders taken during the four quarterly promotion periods in 1 year. Together with the 2% discount mentioned above, the total discounts for preorders amount to 12%. At the same time, there are minimum orders (sales quotas) for the dealers for each promotion period.

^c Computec distributes this list to existing and potential customers so that the latter will solicit business from those that are listed.

^d Although a dealer has to take all deliveries before a specified date, the dealer can divide his total preorder into smaller portions and decide when to take delivery of each portion.

Appendix D: Proof of proposition 1

Suppose R ≥ 2 dealers remain after the exit stage. Denote the preorder of dealer i as q_i and the retail price it subsequently chooses as p_i. Denote the aggregate preorder as $\mathrm{Q}={\displaystyle \sum_{\mathrm{i}=1}^{\mathrm{R}}{\mathrm{q}}_{\mathrm{i}}}$. In this appendix, we prove Proposition 1: If p^Cournot(c;R) > θ, then p = θ and Q = D(θ), whereas if p^Cournot(c;R) ≤ θ, then p = p^Cournot(c;R) and Q = Rq^Cournot(c;R).

Recall that under the “rules of the game,” if the lowest retail price is p, then the D(p) customers arrange themselves so that a lowest-price firm that preordered q_i would get q_iD(p)/Z customers, where Z = ∑_{j∊{lowest price firms}}q_j.

It follows that if D(p)/Z ≤ 1, every dealer has preordered weakly more than its customers demand, and if D(p)/Z > 1, each dealer has preordered less than its customers demand. In the latter case, each dealer is required to satisfy the remaining demand ([D(p)/Z] – 1)q_i > 0) by augmenting the preorder at the undiscounted wholesale price (θ). We denote the inverse demand curve P(·) and assume it is strictly decreasing and weakly concave.

4.1 Pricing subgames

Denote the number of firms remaining after the exit stage as R. Assume R > 1 and that the manufacturer has chosen wholesale price θ and discounted price c. Then, in the symmetric Nash equilibrium of each subgame indexed by {q_i, Q_-i, R} for i = 1,…,R, each dealer will set its retail price equal to p_i = max (0, min(θ, P(Q)). To prove that this profile of retail prices forms a Nash equilibrium in the subgame, we consider three cases:

• Case 1: When observed total preorders are small; that is, P(Q) > θ. In the conjectured equilibrium, p_i = p_-i = θ, Q = D(θ), dealer i would get in revenue θq_i + (θ – θ){[D(θ)/Z] – 1}q_i. The first term is the revenue from selling the preorder. The second term reflects the net margin per unit (its first factor) multiplied by the amount of augmenting required under the rules of the game (the product of the second and third factors). Since consumers would pay the dealer what it cost him to augment (θ per unit), augmenting earns zero margins and the firm’s revenue would be θq_i > 0. Dealer i cannot strictly increase his revenue by unilaterally deviating from p_i = θ. If he strictly raised his retail price, he would lose all his customers and earn zero revenue. Suppose instead he unilaterally reduced the retail price to θ – ε > 0. Then he would earn less on his preorders but would have to augment more to cover all the remaining demand. As a result, he would earn the negative amount (θ – ε) – θ < 0 on every additional unit.

• Case 2: When observed total preorders are large; that is, θ ≥ P(Q) > 0. In the conjectured equilibrium, p_i = p_-i = P(Q), Q = Z. Since D(P(Q))/Z = 1, each dealer’s preorder exactly satisfies the demand of his customers, and there is no need to augment. Dealer i would get in revenue P(Q)q_i ≥ 0. Unilaterally raising his price would result in no customers and no revenue. Unilaterally reducing his price by ε > 0 would result in smaller revenue from his preorders. Moreover, as the firm charging the lowest price, he would be required to augment and would lose money doing so, since he would earn the negative amount (P(Q) – ε) – θ < 0 on each additional unit. Hence, neither deviation would strictly improve dealer i’s revenue.

• Case 3: When observed total preorders are very large; that is, Q ≥ D(0) ⇔ P(Q) = 0. In the conjectured equilibrium, p_i = p_-i = 0. In this case, dealer i would earn zero revenue, but he would also earn nothing if he unilaterally increased his price. Reducing his price unilaterally is infeasible, but if it were not, it would be unprofitable.

  We conclude therefore that if R > 1, every firm will charge the retail price p_i = max(0, min(θ, P(Q)) for i = 1, …, R.

4.2 Preordering subgames

Assume that the number of firms that did not exit is R > 1 and that the manufacturer has chosen wholesale price θ and discounted price c.

• Case 1: Suppose p^Cournot(c;R) > θ. Then in the proposed subgame-perfect equilibrium, q_i = q_-i = D(θ)/R. If this strategy profile is played, every dealer will subsequently set the retail price P(R · D(θ)/R) = θ, since this is the equilibrium price in the pricing subgame that follows. Dealer i would earn q_i D(θ)/R > 0 in the proposed equilibrium. If dealer i unilaterally reduced his preorder, he would incur a strict loss, since the common retail price would remain θ. So he would lose the margin θ – c > 0 on each unit he failed to preorder and would break even on the amount he was required to augment. If dealer i unilaterally increased his preorder instead, his gross margin would also unilaterally decline. To see this, note that his gross margin would equal q_i[P(q_i + (R – 1)D(θ))/R) – c]. Since the first and second derivatives of the inverse demand function are negative by assumption, this function is strictly concave. Moreover, the first derivative of the gross-margin function is strictly negative for q_i > D(θ)/R > q^Cournot(c;R).²³ Finally, if he unilaterally preordered a very large amount, the common retail price in the pricing stage would fall to zero. He would scrap and get zero gross margin from his preorders. Hence, no dealer has a strict incentive to deviate unilaterally in either direction from his proposed equilibrium strategy.

• Case 2: Suppose dealers observe that p^Cournot(c;R) ≤ θ. Then in the proposed equilibrium, q_i = q_-i = q^Cournot(c;R). Since Q = R · q^Cournot(c;R) < D(0) and p^Cournot(c;R) = P(R · q^Cournot(c;R)), the common retail price in the last stage would be 0 < p_i = p_-i = p^Cournot(c;R) ≤ θ. If dealer i deviated locally by preordering a larger amount, the common retail price in the final stage P(Q) would fall to clear the market. Since dealer i would no longer be choosing the Cournot best reply to the unchanged preorders of the other (R – 1) dealers, his net profit would strictly decline. If dealer i unilaterally preordered a very large amount, the common retail price in the pricing stage would fall to zero. Those dealers who had placed a preorder, including him, would scrap; he would get zero gross margin. On the other hand, if he unilaterally preordered an amount smaller than q^Cournot(c;R), the common retail price in the final stage would rise. Suppose it remained below θ. Then dealer i would no longer be choosing the best reply to the preorders of the other (R – 1) dealers, and his net profit would decline. Finally, any unilateral reduction in i’s preorder after the common retail price reached θ would also be harmful. Dealer i would lose the margin θ – c > 0 on each unit reduction in his preorders beyond that point; he would break even on each unit that he was required to augment, since the price customers pay would just cover the undiscounted wholesale price the manufacturer would charge dealer i.

To summarize our findings, in a symmetric subgame-perfect equilibrium, if p^Cournot(c;R) > θ, then p = θ and Q = D(θ), whereas if p^Cournot(c;R) ≤ θ, then p = p^Cournot(c;R) and Q = Rq^Cournot(c;R).

Appendix E: Proof of proposition 2

To analyze the constrained profit-maximization problem of the manufacturer for the R > 1 case, we append the multiplier λ to constraint (1), γ to constraint (2), and δ to constraint (3). The Lagrangean of the problem is

$$ \mathrm{L}=\left(\mathrm{c} - {\textstyle}\mathrm{m}\right)\mathrm{D}\left(\uptheta \right)+\uplambda \left[\mathrm{D}\left(\uptheta \right)\left(\uptheta {\textstyle\ \hbox{--}\ }\mathrm{c}\right) - {\textstyle}\mathrm{R}\mathrm{F}\right]+\upgamma \left[{\mathrm{p}}^{\mathrm{Cournot}}\left(\mathrm{c};\mathrm{R}\right) - {\textstyle}\uptheta \right]+\updelta \left[\mathrm{c} - \mathrm{m}\right]. $$

At an optimum, the following conditions must hold:

$$ \uptheta \ge 0,\left(\mathrm{c}{\textstyle\ \hbox{--}\ }\mathrm{m}\right){\mathrm{D}}^{\prime}\left(\uptheta \right)+\uplambda \left[{\mathrm{D}}^{\prime}\left(\uptheta \right)\left(\uptheta {\textstyle\ \hbox{--}\ }\mathrm{c}\right)+\mathrm{D}\left(\uptheta \right)\right]{\textstyle\ \hbox{--}\ }\upgamma \le 0,\mathrm{c}.\mathrm{s}. $$

(E1)

$$ \mathrm{c}\ge 0,\mathrm{D}\left(\uptheta \right){\textstyle\ \hbox{--}\ }\uplambda \mathrm{D}\left(\uptheta \right)+\upgamma \mathrm{p}{\prime}^{\mathrm{Cournot}}\left(\mathrm{c};\mathrm{R}\right)+\updelta \le 0,\mathrm{c}.\mathrm{s}. $$

(E2)

$$ \uplambda \ge 0,\mathrm{D}\left(\uptheta \right)\left(\uptheta - {\textstyle}\mathrm{c}\right) - {\textstyle}\mathrm{R}\mathrm{F}\ge 0,\mathrm{c}.\mathrm{s}. $$

(E3)

$$ \upgamma \ge 0,{\mathrm{p}}^{\mathrm{Cournot}}\left(\mathrm{c};\mathrm{R}\right) - {\textstyle}\uptheta \ge 0,\mathrm{c}.\mathrm{s}. $$

(E4)

$$ \updelta \ge 0,\mathrm{c}{\textstyle\ \hbox{--}\ }\mathrm{m}\ge 0,\mathrm{c}.\mathrm{s}. $$

(E5)

The notation “c.s.” is an abbreviation for “complementary slackness.” The condition a ≥ 0, b ≥ 0, c.s. means that both a and b are nonnegative and, in addition, that at least one of these variables is zero; thus, the condition eliminates the possibility that both variables are strictly positive.

We begin with three preliminary observations:

• 1. (E2) requires λ > 0, and therefore (E3) requires D(θ)(θ – c) – RF = 0.

• 2. (E3) requires that θ > c > 0. Since θ > 0, (E1) implies (c – m)D'(θ) + λ[D'(θ)(θ – c) + D(θ)] – γ = 0.

• 3. (E5) requires c > 0. Since c > 0, (E2) requires D(θ) – λD(θ) + γp'^Cournot(c;R) + δ = 0.

5.1 Optimum in the interior of the Bertrand region

If the optimum occurs in the interior of the Bertrand region, (E4) requires γ = 0, and since λ > 0, (E3) requires that D(θ)(θ – c) – RF = 0. Moreover, c – m > 0, implying that δ = 0.²⁴ We deduce from (E2) that λ = 1 and from (E1) that θ = θ^*, the implicit solution to (θ – m)D'(θ) + D(θ).

5.2 Optimum on the R-dealer CB boundary of the Bertrand region

On the CB boundary, p^Cournot(c;R) – θ = 0, so (E4) implies that γ ≥ 0. Recall that c^* is the solution to p^Cournot(c;R) – θ^* = 0. That is, (c^*, θ^*) is on the CB boundary at the channel-profit maximizing wholesale price. If this is not a solution, it is because D(p^Cournot(c^*;R))(p^Cournot(c^*;R) – c^*) – RF < 0. The left-hand side is the sum of dealer profits in Cournot equilibrium where each firm has constant marginal cost c = c^*. It has been shown (Kotchen and Salant 2010, p. 247) that, given our assumption that the inverse demand curve is weakly concave, Cournot industry profits are strictly decreasing in the common constant marginal cost. It follows that if D(p^Cournot(c^*;R))(p^Cournot(c^*;R) – c^*) – RF < 0, the constrained optimum occurs at θ < θ^*.

5.3 Existence and uniqueness

If RF is sufficiently large, the constraint set will be empty and the manufacturer goes out of business, earning a payoff of zero. If the constraint set is nonempty, the solution to the constrained maximization problem must be unique. Recall that this solution occurs either in the interior of the Bertrand region with θ = θ^* or on the R-dealer CB boundary. We first prove that if there exists one solution in the interior of the Bertrand region (respectively, on the R-dealer CB boundary), there cannot be any other solution in the interior of the Bertrand region (respectively, on the R-dealer CB boundary). We conclude by proving that if a solution occurs in one of these two regions, there cannot be an additional solution in the other region.

There can be at most one solution in the interior of the Bertrand region. The assumption that the inverse demand curve is weakly concave ensures that channel profits are a strictly concave function of θ, and hence there is a unique channel-profit maximizing wholesale price. Given that price, if one discounted price (c) generates dealer gross margins exactly equal to their fixed costs, no other discounted price will achieve exactly those margins, since dealer gross margins monotonically fall as the preorder cost rises.

There can also be at most one solution on the R-dealer CB boundary, for along that boundary, dealer gross margins increase monotonically as c decreases.

The optimum can occur in only one of these two regions, because as one reduces c and moves horizontally across the Bertrand region at θ^* and then continues down the R-dealer CB boundary (θ < θ^*), dealer gross margins monotonically increase. Hence, if they exactly covered fixed costs in the interior of the Bertrand region, they must exceed the fixed costs anywhere on the R-dealer CB boundary. Similarly, if they exactly covered the fixed costs at θ < θ^* somewhere on the R-dealer CB boundary, they must be strictly smaller at any point in the interior of the Bertrand region with the wholesale price set at the channel-profit maximizing price (θ^*).

Appendix F: Mathematical proof of example

6.1 The case of R = 1

We begin by deriving the profit function of the manufacturer when demand is linear and the one remaining dealer has fixed cost F.

Given D(p) = (a – p) / b, in the Spengler’s (1950) case, the single dealer’s optimal retail price for a given manufacturer’s wholesale price θ is

$$ \mathrm{p}\left(\uptheta \right)= \arg { \max}_{\mathrm{p}\ge 0}\left(\mathrm{p} - \uptheta \right)\frac{\left(\mathrm{a} - \mathrm{p}\right)}{\mathrm{b}}-\mathrm{F}=\frac{\mathrm{a}+\uptheta}{2}. $$

And the single dealer’s optimal quantity is D(p(θ)) = $\frac{\mathrm{a} - \uptheta}{2\mathrm{b}}$. The manufacturer maximizes his profit, $\left(\uptheta - \mathrm{m}\right)\frac{\left(\mathrm{a} - \uptheta \right)}{2\mathrm{b}}$, and obtains his optimal wholesale price θ^* = (a + m) / 2 . Maximized manufacturer profit is therefore

$$ {\uppi}_{\mathrm{m}}^1=\left(\frac{\mathrm{a}+\mathrm{m}}{2} - \mathrm{m}\right)\left(\frac{\mathrm{a} - \mathrm{m}}{4\mathrm{b}}\right)=\frac{{\left(\mathrm{a} - \mathrm{m}\right)}^2}{8\mathrm{b}}=\frac{1}{2}{\Pi}_{\max }. $$

(F1)

The Spengler solution holds provided the gross margin of the dealer weakly exceeds his fixed cost. Substituting the optimal wholesale price θ^* = (a + m) / 2 into the exclusive dealer’s profit function, we get

${\uppi}_{\mathrm{d}}=\left(\mathrm{p}\left(\uptheta \right)\ \hbox{--}\ \uptheta \right)\mathrm{D}\left(\mathrm{p}\left(\uptheta \right)\right)\ \hbox{--}\ \mathrm{F}=\left(\frac{\mathrm{a} + \uptheta}{2} - \uptheta \right)\left(\frac{\mathrm{a} - \uptheta}{2\mathrm{b}}\right) - \mathrm{F}=\frac{{\left(\mathrm{a} - \mathrm{m}\right)}^2}{16\mathrm{b}} - \mathrm{F}$. Thus, Spengler’s solution holds for any $\mathrm{F}\le \frac{{\left(\mathrm{a} - \mathrm{m}\right)}^2}{16\mathrm{b}}=\frac{1}{4}{\Pi}_{\max }.$

If the fixed cost is larger, however, the manufacturer must reduce his wholesale price or his single dealer will exit. In this case the endogenous variables (p, θ, π’s) are determined by the two constraints (4) and (5) in the main text and by the following equation defining manufacturer profit:

$$ {\uppi}_{\mathrm{m}}^1=\left(\uptheta - \mathrm{m}\right)\mathrm{D}\left(\mathrm{p}\left(\uptheta \right)\right)=\left(\uptheta - \mathrm{m}\right)\frac{\left(\mathrm{a} - \uptheta \right)}{2\mathrm{b}}=0. $$

Hence, $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^1}{\mathrm{d}\uptheta}=\frac{\mathrm{a}+\mathrm{m} - 20}{2\mathrm{b}}$. On the other hand, substituting $\mathrm{p}\left(\uptheta \right)=\frac{\mathrm{a} + \uptheta}{2}$ and $\mathrm{D}\left(\mathrm{p}\left(\uptheta \right)\right)=\frac{\mathrm{a} - \uptheta}{2\mathrm{b}}$ into (5) and differentiating, we obtain $\frac{\mathrm{d}\mathrm{F}}{\mathrm{d}\uptheta}=\frac{-\left(\mathrm{a} - \uptheta \right)}{2\mathrm{b}}<0$. Therefore,

$$ \frac{\mathrm{d}{\uppi}_{\mathrm{m}}^1}{\mathrm{d}\mathrm{F}}=\frac{\uptheta - \mathrm{m}}{\mathrm{a}-\uptheta}-1. $$

(F2)

Equation (F2) implies that the second portion of π ¹_m starts to descend at $\mathrm{F}=\frac{\prod_{\max }}{4}$ and since θ^* = (a + m) / 2, $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^1}{\mathrm{d}\mathrm{F}}=0$, implying that there is no kink in π ¹_m at the transition point. Moreover, $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^1}{\mathrm{d}\mathrm{F}}$ is larger than −1 because the first term in (F2) is positive; but since $\frac{\mathrm{d}\mathrm{F}}{\mathrm{d}\uptheta}<0$, this term is decreasing in F. Hence, the second portion of π ¹_m starts out flat and is decreasing and strictly concave in F until it reaches the horizontal axis (π ¹_m  = 0). Since at that point θ = m, Eq. (F3) implies $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^1}{\mathrm{d}\mathrm{F}}=-1$.

6.2 The case of R = 2

In the text, we show that in the Bertrand region the manufacturer’s profit function π ²_m  = П_max − 2F declines linearly until the transition point derived in the text. Here, we describe the profit function, π ²_m , if F is larger, and hence the manufacturer operates in the Cournot region.

In the Cournot region, the common retail price is p^Cournot(c; 2) = $\frac{\mathrm{a}+2\mathrm{c}}{3}$, and each dealer preorders and sells D(p^Cournot(c; 2)) / 2 = $\frac{\mathrm{a} - \mathrm{c}}{3\mathrm{b}}$. Since dealers’ total gross margins just their fixed costs, (p^Cournot(c; 2) – c)D(p^Cournot(c; 2)) = 2(a – c)²/(9b) = 2 F. Solving for F, we conclude that F = (a – c)²/(9b). Thus a one dollar increase in fixed costs requires the manufacturer to cut the preorder cost at a rate of $\frac{\mathrm{dc}}{\mathrm{dF}}=-\frac{9\mathrm{b}}{2\left(\mathrm{a} - \mathrm{c}\right)}<0$. Since the manufacturer’s profit is

$$ {\uppi}_{\mathrm{m}}^2=\left(\mathrm{c} - \mathrm{m}\right)\mathrm{D}\left({\mathrm{p}}^{\mathrm{Cournot}}\left(\mathrm{c}\right)\right)=\frac{2\left(\mathrm{a} - \mathrm{c}\right)\left(\mathrm{c} - \mathrm{m}\right)}{3\mathrm{b}}, $$

we have $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^2}{\mathrm{d}\mathrm{c}}=\frac{2\left(\mathrm{a} + \mathrm{m} - 2\mathrm{c}\right)}{3\mathrm{b}}$. This is negative because c < θ^* = (a + m) / 2. Using these results, we have

$$ \frac{\mathrm{d}{\uppi}_{\mathrm{m}}^2}{\mathrm{d}\mathrm{F}}=\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^2}{\mathrm{d}\mathrm{c}}\cdot \frac{\mathrm{d}\mathrm{c}}{\mathrm{d}\mathrm{F}}=-\frac{3\left(\mathrm{a} - 2\mathrm{c}+\mathrm{m}\right)}{\mathrm{a} - \mathrm{c}}=\frac{\mathrm{a} - \mathrm{m}}{{\left(\mathrm{b}\mathrm{F}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} - 6. $$

(F3)

The last equality uses the fact that $\mathrm{c}=\mathrm{a} - 3{\left(\mathrm{b}\mathrm{F}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ since $\mathrm{F}=\frac{{\left(\mathrm{a} - \mathrm{c}\right)}^2}{9\mathrm{b}}$. Since $\frac{\mathrm{d}{\uppi}_{\mathrm{m}}^2}{\mathrm{d}\mathrm{F}}$ in (F3) is the product of a positive derivative and a negative derivative, it is negative. Since the first term on the right-hand side of (F3) is positive and decreasing in F, the manufacturer’s profit function in this region is not only strictly decreasing but also strictly concave.